Testing the Addition of a New Anti-Cancer Drug, Glofitamab to Usual Chemotherapy for Burkitt and Double Hit Lymphoma

Inclusion criteria

* BURKITT LYMPHOMA (BL) COHORT 1: Histologically confirmed Burkitt lymphoma by International Consensus Classification criteria. Patients with Burkitt lymphoma must have one or more of the following adverse risk factors at diagnosis: * Stage III or IV disease * Elevated lactate dehydrogenase (LDH) greater than institutional upper limit of normal (ULN) * Tumor mass ≥ 7 cm * BL COHORT 1: No prior treatment except for pre-phase chemotherapy, pre-phase corticosteroids, or one cycle of chemotherapy as described below. * Pre-phase chemotherapy with corticosteroids or cyclophosphamide/prednisone is permitted prior to enrollment. * One cycle of prior chemotherapy (for example, cyclophosphamide, doxorubicin, vincristine and prednisone \[CHOP\], Polatuzumab-cyclophosphamide doxorubicin and prednisone \[CHP\], cyclophosphamide, vincristine, doxorubicin and methotrexate \[CODOXM\], or etoposide, prednisone, vincristine, cyclophosphamide and doxorubicin \[EPOCH\] \[with or without rituximab\]) may be administered no more than 21 days prior to enrollment. * Patients enrolling on study after one cycle of prior anthracycline containing therapy will start the study therapy in cycle 2 and cycle 2 must occur no later than 22-36 days after the start of cycle 1. (If feasible, cycle 2 should start on day 22, but +14 days is permitted to permit recovery from toxicity in cycle 1) * BL COHORT 1: Age ≥ 18 years * BL COHORT 1: Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2. ECOG performance status of 3 is permitted if related to lymphoma * BL COHORT 1: Absolute neutrophil count (ANC) ≥ 1,000/mm\^3 (unless attributable to lymphoma) * BL COHORT 1: Platelet count ≥ 75,000/mm\^3 (unless attributable to lymphoma) * BL COHORT 1: Calculated (Calc.) creatinine clearance ≥ 50 mL/min (unless attributable to lymphoma) * BL COHORT 1: Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\])/ alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) ≤ 3 x institutional ULN (unless attributable to lymphoma) * BL COHORT 1: Total bilirubin ≤ 2.0 mg/dL (unless due to Gilbert's syndrome) (unless attributable to lymphoma) * BL COHORT 1: International normalization ratio (INR) OR prothrombin time (PT) \> 1.5 x institutional ULN (unless attributable to lymphoma) * BL COHORT 1: Partial thromboplastin time (PTT) or activated partial thromboplastin time (aPTT) \> 1.5 x institutional ULN (unless attributable to lymphoma) * BL COHORT 1: Not pregnant and not nursing, because this study involves agents that have known genotoxic, mutagenic and teratogenic effects. Therefore, for women of childbearing potential only, a negative pregnancy test done ≤ 28 days prior to registration is required * BL COHORT 1: Patients with prior malignancy or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen (i.e. anthracycline dosing that will exceed recommended lifetime maximum dosing) are eligible for this trial * BL COHORT 1: Patients with HIV are eligible but must be on effective anti-retroviral therapy, with undetectable viral load within 6 months of enrollment * BL COHORT 1: Patients with a history of hepatitis B who are hepatitis B (HepB) surface antigen (Ag) positive and/or HepB core antibody (Ab) positive with undetectable Hep B viral load who start suppressive antiviral therapy prior to chemotherapy are eligible * BL COHORT 1: Patients with a history of hepatitis C infection that have been treated for hepatitis C virus (HCV) and have an undetectable HCV viral load are eligible * BL COHORT 1: No active ischemic heart disease or congestive heart failure, AND adequate cardiac function defined as ejection fraction of ≥ 45% * BL COHORT 1: Patients with CSF/leptomeningeal involvement at diagnosis are eligible. Patients with parenchymal brain involvement at diagnosis are NOT eligible * BL COHORT 1: Patients requiring \> 10 mg/day of prednisone or equivalent, for the treatment of conditions other than the control of lymphoma related symptoms, are NOT eligible * BL COHORT 1: Patients with major surgery within 4 weeks other than for diagnosis of lymphoma are NOT eligible * BL COHORT 1: Patients with known active infections, or reactivation of a latent infection (excluding fungal infections of nail bed, hepatitis B on suppressive therapy or treated hepatitis C) requiring hospitalization or treatment with IV antibiotics within 4 weeks of study treatment are NOT eligible * BL COHORT 1: Patients with a prior history of hemophagocytic lymphohistocytosis (HLH) are NOT eligible * BL COHORT 1: Patients with a history of autoimmune disease (including but not limited to myasthenia gravis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, Sjogren's syndrome, Guillain-Barre syndrome, or multiple sclerosis) are NOT eligible. However, patients with autoimmune related hypothyroidism on a stable dose of thyroid replacement hormone, patients with a history of lymphoma related immune thrombocytopenic purpura or hemolytic anemia, and patients with type 1 or type 2 diabetes on a stable insulin regimen are eligible to enroll. In addition, patients with eczema, lichen simplex chronicus, or vitiligo with dermatologic manifestations only may also enroll if the rash covers \< 10% of body surface area and the disease is well controlled and requires only topical corticosteroids. Lastly, patients treated with prior immunotherapeutic agents with a history of any grade 3 events (except for grade 3 endocrinopathy managed with replacement therapy) or grade 1-2 events that did not resolve to baseline with treatment discontinuation are NOT eligible. * BL COHORT 1: Patients with a history of progressive multifocal leukoencephalopathy (PML) are NOT eligible * BL COHORT 1: Patients who have received live, attenuated vaccines within 4 weeks prior to cycle 1 day (D) 1 of study treatment are NOT eligible * DOUBLE HIT LYMPHOMA (DHL) COHORT 2: Histologically confirmed high grade/double hit B-cell lymphoma with MYC and BCL2 translocations by International Consensus Classification (ICC) or World Health Organization (WHO) criteria. Fluorescence in situ hybridization (FISH) For MYC and BCL2 translocations must be performed by the referring site and must be positive for translocations of BOTH MYC and BCL2. Patients found to have BCL6 translocations in addition to BOTH MYC and BCL2 (so called "triple hit lymphoma") are eligible * DHL COHORT 2: Stage II-IV disease per Lugano staging classification * DHL COHORT 2: No prior treatment except for pre-phase chemotherapy, pre-phase corticosteroids, or one cycle of chemotherapy as described below. * Pre-phase chemotherapy with corticosteroids or cyclophosphamide/prednisone is permitted prior to enrollment. * One cycle of prior chemotherapy (for example, CHOP, Polatuzumab-CHP, or EPOCH (with or without Rituximab)) may be administered no more than 21 days prior to enrollment. * Patients enrolling on study after one cycle of prior anthracycline containing therapy will start the study therapy in cycle 2 and cycle 2 must occur no later than 22-36 days after the start of cycle 1. (If feasible, cycle 2 should start on day 22, but +14 days is permitted to permit recovery from toxicity in cycle 1) * DHL COHORT 2: Age ≥ 18 years * DHL COHORT 2: ECOG performance status ≤ 2. ECOG performance status of 3 is permitted if related to lymphoma * DHL COHORT 2: Absolute neutrophil count (ANC) ≥ 1,000/mm\^ 3 (unless attributable to lymphoma) * DHL COHORT 2: Platelet Count ≥ 75,000/mm\^3 (unless attributable to lymphoma) * DHL COHORT 2: Calc. Creatinine Clearance ≥ 50 mL/min (unless attributable to lymphoma) * DHL COHORT 2: AST(SGOT)/ALT(SGPT) ≤ 3 x institutional ULN (unless attributable to lymphoma) * DHL COHORT 2: Total Bilirubin ≤ 2.0 mg/dL (unless due to Gilbert's Syndrome) (unless attributable to lymphoma) * DHL COHORT 2: INR OR PT \> 1.5 x institutional ULN (unless attributable to lymphoma) * DHL COHORT 2: PTT or aPTT \> 1.5 x institutional ULN (unless attributable to lymphoma) * DHL COHORT 2: Not pregnant and not nursing, because this study involves agents that have known genotoxic, mutagenic and teratogenic effects. Therefore, for women of childbearing potential only, a negative pregnancy test done ≤ 28 days prior to registration is required * DHL COHORT 2: Patients with prior malignancy or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen (i.e. anthracycline dosing that will exceed recommended lifetime maximum dosing) are eligible for this trial * DHL COHORT 2: Patients with HIV are eligible but must be on effective anti-retroviral therapy, with undetectable viral load within 6 months of enrollment * DHL COHORT 2: Patients with a history of hepatitis B who are HepB surface Ag positive and/or HepB core Ab positive with undetectable Hep B viral load who who start suppressive antiviral therapy prior to chemotherapy are eligible * DHL COHORT 2: Patients with a history of hepatitis C infection that have been treated for HCV and have an undetectable HCV viral load are eligible * DHL COHORT 2: No active ischemic heart disease or congestive heart failure, AND adequate cardiac function defined as ejection fraction of ≥ 45% * DHL COHORT 2: Patients with CSF/leptomeningeal involvement at diagnosis are eligible. Patients with parenchymal brain involvement at diagnosis are NOT eligible * DHL COHORT 2: Corticosteroid use: Patients requiring \> 10 mg/day of prednisone or equivalent, for the treatment of conditions other than the control of lymphoma related symptoms, are NOT eligible * DHL COHORT 2: Patients with major surgery within 4 weeks other than for diagnosis of lymphoma are NOT eligible * DHL COHORT 2: Patients with known active infections, or reactivation of a latent infection (excluding fungal infections of nail bed, hepatitis B on suppressive therapy, or treated hepatitis C) requiring hospitalization or treatment with IV antibiotics within 4 weeks of study treatment are NOT eligible * DHL COHORT 2: Patients with a prior history of HLH are NOT eligible * DHL COHORT 2: Patients with a history of autoimmune disease (including but not limited to myasthenia gravis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, Sjogren's syndrome, Guillain-Barre syndrome, or multiple sclerosis) are NOT eligible. However, patients with autoimmune related hypothyroidism on a stable dose of thyroid replacement hormone, patients with a history of lymphoma related immune thrombocytopenic purpura or hemolytic anemia, and patients with type 1 or type 2 diabetes on a stable insulin regimen are eligible to enroll. In addition, patients with eczema, lichen simplex chronicus, or vitiligo with dermatologic manifestations only may also enroll if the rash covers \< 10% of body surface area and the disease is well controlled and requires only topical corticosteroids. Lastly, patients treated with prior immunotherapeutic agents with a history of any grade 3 events (except for grade 3 endocrinopathy managed with replacement therapy) or grade 1-2 events that did not resolve to baseline with treatment discontinuation are NOT eligible * DHL COHORT 2: Patients with a history of PML are NOT eligible * DHL COHORT 2: Patients who have received live, attenuated vaccines within 4 weeks prior to cycle 1 D1 of study treatment are NOT eligible * RELAPSED BL AND HGL COHORT 3: Histologically confirmed Burkitt lymphoma by International Consensus Classification criteria * RELAPSED BL AND HGL COHORT 3: Patients must have relapsed or refractory disease after at least one prior treatment with chemotherapy or chemoimmunotherapy. * Prior radiotherapy in addition or in conjunction with prior chemotherapy is permitted. * Patients who have previously received a bispecific antibody (BsAb) are NOT eligible. However, patients enrolled to the standard arms on cohorts 1 and 2 of this trial who did not receive glofitimab and have relapsed or refractory disease are eligible to enroll on cohort 3. * Prior treatment with systemic immunotherapeutic agents including antibody drug conjugates, radio-immunoconjugates, or immune/cytokine direct therapy must have occurred within 3 weeks or five half-lives of the drug, whichever is shorter * RELAPSED BL AND HGL COHORT 3: Age ≥ 18 years * RELAPSED BL AND HGL COHORT 3: ECOG Performance Status ≤ 2. ECOG performance status of 3 is permitted if related to lymphoma * RELAPSED BL AND HGL COHORT 3: Absolute Neutrophil Count (ANC) ≥ 1,000/mm\^3 (unless attributable to lymphoma) * RELAPSED BL AND HGL COHORT 3: Platelet Count ≥ 75,000/mm\^3 (unless attributable to lymphoma) * RELAPSED BL AND HGL COHORT 3: Calc. Creatinine Clearance ≥ 50 mL/min (unless attributable to lymphoma) * RELAPSED BL AND HGL COHORT 3: AST(SGOT)/ALT(SGPT) ≤ 3 x institutional ULN (unless attributable to lymphoma) * RELAPSED BL AND HGL COHORT 3: Total bilirubin ≤ 2.0 mg/dL (unless due to Gilbert's Syndrome) (unless attributable to lymphoma) * RELAPSED BL AND HGL COHORT 3: INR OR PT \> 1.5 x institutional ULN (unless due to Gilbert's Syndrome) (unless attributable to lymphoma) * RELAPSED BL AND HGL COHORT 3: PTT or aPTT \> 1.5 x institutional ULN (unless due to Gilbert's Syndrome) (unless attributable to lymphoma) * RELAPSED BL AND HGL COHORT 3: Not pregnant and not nursing, because this study involves agents that have known genotoxic, mutagenic and teratogenic effects. Therefore, for women of childbearing potential only, a negative pregnancy test done ≤ 28 days prior to registration is required * RELAPSED BL AND HGL COHORT 3: Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen (i.e. anthracycline dosing that will exceed recommended lifetime maximum dosing) are eligible for this trial * RELAPSED BL AND HGL COHORT 3: Patients with HIV are eligible but must be on effective anti-retroviral therapy, with undetectable viral load within 6 months of enrollment * RELAPSED BL AND HGL COHORT 3: Patients with a history of hepatitis B who are HepB surface Ag positive and/or HepB core Ab positive with undetectable Hep B viral load who are taking suppressive antiviral therapy are eligible * RELAPSED BL AND HGL COHORT 3: Patients with a history of hepatitis C infection that have been treated for HCV and have an undetectable HCV viral load are eligible * RELAPSED BL AND HGL COHORT 3: No active ischemic heart disease or congestive heart failure, AND adequate cardiac function defined as ejection fraction of ≥ 45% * RELAPSED BL AND HGL COHORT 3: Patients with CSF/leptomeningeal involvement at diagnosis are eligible. Patients with parenchymal brain involvement at diagnosis are NOT eligible * RELAPSED BL AND HGL COHORT 3: Corticosteroid use: Patients requiring \> 10 mg/day of prednisone or equivalent, for the treatment of conditions other than the control of lymphoma related symptoms, are NOT eligible * RELAPSED BL AND HGL COHORT 3: Patients with major surgery within 4 weeks other than for diagnosis of lymphoma are NOT eligible * RELAPSED BL AND HGL COHORT 3: Patients with known active infections, or reactivation of a latent infection (excluding fungal infections of nail bed, hepatitis B on suppressive therapy, or treated hepatitis C) requiring hospitalization or treatment with IV antibiotics within 4 weeks of study treatment are NOT eligible * RELAPSED BL AND HGL COHORT 3: Patients with a prior history of HLH are NOT eligible * RELAPSED BL AND HGL COHORT 3: Patients with a history of autoimmune disease (including but not limited to myasthenia gravis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, Sjogren's syndrome, Guillain-Barre syndrome, or multiple sclerosis) are NOT eligible. However, patients with autoimmune related hypothyroidism on a stable dose of thyroid replacement hormone, patients with a history of lymphoma related immune thrombocytopenic purpura or hemolytic anemia, and patients with type 1 diabetes on a stable insulin regimen are eligible to enroll. In addition, patients with eczema, lichen simplex chronicus, or vitiligo with dermatologic manifestations only may also enroll if the rash covers \< 10% of body surface area and the disease is well controlled and requires only topical corticosteroids. Lastly, patients treated with prior immunotherapeutic agents with a history of any grade 3 events (except for grade 3 endocrinopathy managed with replacement therapy) or grade 1-2 events that did not resolve to baseline with treatment discontinuation are NOT eligible * RELAPSED BL AND HGL COHORT 3: Patients with a history of PML are NOT eligible * RELAPSED BL AND HGL COHORT 3: Patients who have received live, attenuated vaccines within 4 weeks prior to cycle 1 D1 of study treatment are NOT eligible