Glofitamab With Obinutuzumab Pre-treatment for the Treatment of Central Nervous System Lymphoma

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative * Assent, when appropriate, will be obtained per institutional guidelines * Agreement to allow the use of archival tissue from diagnostic tumor biopsies * If unavailable, exceptions may be granted with study principal investigator (PI) approval * Age: ≥ 18 years * Karnofsky performance status (KPS) ≥ 30 * Histologically confirmed primary or secondary CNS lymphoma. Tumor must be positive for CD20 by immunohistochemistry or flow cytometry on the most recent biopsy. Neuroimaging alone is acceptable in secondary CNS lymphoma cases where all of the following criteria are met: 1) brain MRI findings are consistent with CNS lymphoma, 2) the disease has been histologically documented in other sites, 3) the CNS lesions are concomitant with systemic progression, and 4) a brain biopsy would be unadvised per the treating provider. * Cohort 1: Primary CNS lymphoma * Cohort 2: Secondary CNS lymphoma * Patients must not require urgent treatment initiation due to bulky or rapidly progressing CNS lymphoma that poses risk for impending critical brain failure. This includes \> 5 mm of midline shift, radiographic evidence of impending brain herniation, or clinical evidence of significantly increased intracranial pressure such as papilledema * Have failed methotrexate-based therapy or are ineligible/refuse high-dose methotrexate treatment (e.g. creatinine clearance \[CrCl\] \< 50 mL/min, effusions, ascites etc) * Measurable CNS lymphoma based on gadolinium enhancement of brain or spine MRI and/or positive CSF or intravitreal fluid cytology * Fully recovered from the acute toxic effects (except alopecia) to ≤ grade 1 to prior anti-cancer therapy * Alert and able to participate in a full neurological exam * Without bone marrow involvement: Absolute neutrophil count (ANC) ≥ 1,000/mm\^3 * NOTE: Growth factor is not permitted within 14 days of ANC assessment unless cytopenia is secondary to disease involvement * With bone marrow involvement: ANC ≥ 500/mm\^3 * NOTE: Growth factor is not permitted within 14 days of ANC assessment unless cytopenia is secondary to disease involvement * Without bone marrow involvement: Platelets ≥ 75,000/mm\^3 * NOTE: Platelet transfusions are not permitted within 14 days of platelet assessment unless cytopenia is secondary to disease involvement * With bone marrow involvement: Platelets ≥ 50,000/mm\^3 * NOTE: Platelet transfusions are not permitted within 14 days of platelet assessment unless cytopenia is secondary to disease involvement * Hemoglobin ≥ 8.0 g/dL * Total bilirubin ≤ 1.5 x upper limit of normal (ULN) (unless has Gilbert's disease, then ≤ 3.0 x ULN is allowed) * Aspartate aminotransferase (AST) ≤ 2.5 x ULN * Alanine aminotransferase (ALT) ≤ 2.5 x ULN * Creatinine clearance of ≥ 30 mL/min per 24 hour urine test or the Cockcroft-Gault formula * If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin (PT) ≤ 1.5 x ULN * If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants * If not receiving anticoagulants: Activated partial thromboplastin time (aPTT) ≤ 1.5 x ULN * If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants * Seronegative for HIV antigen/antibody (Ag/Ab) combo * Individuals with a positive HIV test at screening are eligible provided, prior to enrollment, they are stable on antiretroviral therapy, have a CD4 count ≥ 200/uL, and have an undetectable viral load * Seronegative for active hepatitis B virus (HBV) (surface antigen negative) * Participants with occult or prior hepatitis B infection (defined as positive total hepatitis B core antibody and negative hepatitis B virus surface antigen \[HBsAg\]) may be included if hepatitis B virus (HBV) deoxyribonucleic acid (DNA) is undetectable at the time of screening. Such participants must be willing to undergo HBV DNA testing on day 1 of every cycle and every 3 months for at least 12 months after the final cycle of study treatment and appropriate antiviral therapy as indicated * Seronegative for HCV * Participants positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA) * Women of childbearing potential (WOCBP): Negative serum pregnancy test * Agreement by females and males of childbearing potential to use an effective method of birth control or abstain from heterosexual activity for the course of the study and after completion of study treatment as described below separately for males and females * Female participants must remain abstinent or use contraceptive methods with a failure rate of \< 1% per year during the treatment period and for at least 18 months after pretreatment with obinutuzumab, 2 months after the final dose of glofitamab and 3 months after the final dose of tocilizumab (as applicable), whichever is longer. Women must refrain from donating eggs during this same period * Examples of contraceptive methods with a failure rate of \< 1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices * Hormonal contraceptive methods must be supplemented by a barrier method * For male participants: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods, and agree to refrain from donating sperm, as defined below: * With a female partner of childbearing potential or pregnant female partners, male participants must remain abstinent or use a condom plus an additional contraceptive method that together result in a failure rate of \< 1% per year during the treatment period and for at least 3 months after pretreatment with obinutuzumab, 2 months after the final dose of glofitamab or 2 months after the last dose of tocilizumab (as applicable), whichever is longer. Male participants must refrain from donating sperm during this same period * The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the individual. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not adequate methods of contraception. * Childbearing potential defined as not being permanently surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only). Per this definition, a female participant with tubal ligation is considered to be of childbearing potential Exclusion Criteria: * Contraindication to any of the individual components of glofitamab or history of severe allergic or anaphylactic reactions to humanized or murine monoclonal antibodies or known sensitivity or allergy to murine products * Prior solid organ transplantation * Prior treatment with systemic immunotherapeutic agents, including but not limited to, radio-immuno-conjugates, antibody-drug conjugates, immune/cytokines and monoclonal antibodies (mAbs) (e.g., anti cytotoxic T lymphocyte associated protein 4, anti PD-1, and anti PD-L1) within 2 weeks or five half-lives of the drug, whichever is shorter, prior to day 1 of protocol therapy * Prior treatment with glofitamab or other CD20 x CD3 bispecific antibodies * Prior use of systemic chemotherapy within 2 weeks of the start of cycle 1 * Prior treatment with intrathecal chemotherapy within 1 week of the start of cycle 1. Note, in patients with lymphoma restricted to the CSF and no other measurable sites of CNS disease, positive CSF cytology must be documented following the most recent administration of intrathecal chemotherapy * Prior treatment with radiotherapy within 2 weeks prior to day 1 of protocol therapy * If patients have received radiotherapy within 4 weeks prior to day 1 of protocol therapy, patients must have at least one measurable lesion outside of the radiation field. Patients who have only one measurable lesion that was previously irradiated but subsequently progressed are eligible * Prior treatment with chimeric antigen receptor T cell (CAR-T) therapy within 30 days prior to day 1 of protocol therapy * Any investigational therapy for the purposes of treating cancer within 21 days prior to the start of cycle 1 * Corticosteroid use for purposes other than lymphoma symptom control * The use of inhaled corticosteroids is permitted * The use of mineralocorticoids for management of orthostatic hypotension is permitted. * The use of physiologic doses of corticosteroids for management of adrenal insufficiency is permitted. * Participants who require lymphoma symptom control during screening may receive steroids in the following manner: * Up to 50 mg/day of prednisone or equivalent may be used for lymphoma symptom control during screening * History of other malignancy that could affect compliance with the protocol or interpretation of results: * Participants with a history of curatively treated basal or squamous cell carcinoma or melanoma of the skin or in situ carcinoma of the cervix at any time prior to the study are eligible * Participants with any malignancy appropriately treated with curative intent and the malignancy has been in remission without treatment for \> 2 years prior to enrollment are eligible * Participants with low-grade, early-stage prostate cancer (Gleason score 6 or below, stage 1 or 2) with no requirement for therapy at any time prior to study are eligible * If the malignancy is expected to not require any treatment for at least 2 years (this exception should be discussed with the study PI) * Unstable cardiac disease as defined by one of the following: * Cardiac events such as myocardial infarction (MI) within the past 3 months * NYHA (New York Heart Association) heart failure class III-IV * Unstable arrhythmias, or unstable angina * Recent major surgery within 4 weeks prior to the start of cycle 1, other than for diagnosis * Current or past history of CNS disease, such as stroke, epilepsy, CNS vasculitis or neurodegenerative disease with the exception of CNS lymphoma. Exceptions can be granted after discussion with the PI of the study * Participants with a history of stroke who have not experienced a stroke or transient ischemic attack in the past 2 years and have no residual neurological deficits, as judged by the investigator, are allowed * Known or suspected history of hemophagocytic lymphohistiocytosis (HLH) * History or presence of an abnormal electrocardiogram (ECG) that is clinically significant in the investigator's opinion * Known or suspected active infection, or reactivation of a latent infection, whether bacterial, viral, fungal, mycobacterial, or other pathogens (excluding fungal infections of nail beds) or any major episode of infection requiring hospitalization or treatment with IV antibiotics (for IV antibiotics this pertains to completion of last course of antibiotic treatment) within 4 weeks of dosing * Active autoimmune disease requiring treatment with immune suppressive medications other than physiologic doses of steroids * Participants with a history of autoimmune-related hypothyroidism on a stable dose of thyroid-replacement hormone may be eligible * Participants with controlled type 1 diabetes mellitus who are on an insulin regimen are eligible for the study * Participants with eczema, psoriasis, lichen simplex chronicus, or vitiligo with dermatologic manifestations only are eligible (e.g., participants with psoriatic arthritis are excluded) if all the following conditions are met: * Rash covers \< 10% of body surface area * Disease is well controlled for the last 12 months and requires only low potency topical corticosteroids * Clinically significant liver disease, including active viral or other hepatitis or cirrhosis * Live, attenuated vaccine within 4 weeks before study treatment infusion on day 1 of cycle 1 or anticipation that such a live, attenuated vaccine will be required during the study. Live vaccines during the study and until participants B cells recover, are prohibited * Influenza vaccination should be given during influenza season only. Participants must not receive live, attenuated influenza vaccine at any time during the study treatment period * Suspected active or latent tuberculosis (as confirmed by a positive interferon gamma release assay) * Participants with a history of progressive multifocal leukoencephalopathy * Females only: Pregnant or breastfeeding or intention of becoming pregnant during the study or within 18 months after pretreatment with obinutuzumab or 2 months after the final dose of glofitamab, whichever is longer * Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)