Allogeneic CD6 Chimeric Antigen Receptor T Regulatory Cells (CD6-CAR Tregs) for the Treatment of Patients With Chronic Graft Versus Host Disease After Allogeneic Hematopoietic Cell Transplantation

Inclusion Criteria: * PATIENT (RECIPIENT) INCLUSION CRITERIA: * All participants must have the ability to understand and the willingness to sign a written informed consent * Participants must agree to allow the use of archival tissue from diagnostic biopsies. * If unavailable, exceptions may be granted with study PI approval Note: For research participants who do not speak English, a short form consent may be used with a City of Hope (COH) certified interpreter/translator to proceed with screening and leukapheresis, while the request for a translated full consent is processed * Age \>= 18 years * Karnofsky performance status of \>= 70% * Received allogeneic hematopoietic stem cell transplantation (alloHCT) from matched related or haploidentical donor as part of treatment of hematologic disorders Note: The donor needs to consent for leukapheresis * Clinical diagnosis of steroid-dependent or refractory, moderate to severe cGVHD * Steroid refractory cGVHD defined as having persistent signs and symptoms of cGVHD per institutional policy despite the use of prednisone for 2 months without complete resolution of signs and symptoms * Estimated life expectancy \> 90 days * Stable dose of corticosteroids for \>= 14 days prior to enrollment not exceeding 15mg/day of prednisone or equivalent + with up to 7ng/mL/day sirolimus with therapeutic drug monitoring * Exposure to at least 1 of the Food and Drug Administration (FDA) approved tyrosine kinase inhibitor (TKI) therapies for cGVHD * Naive to anti-CD6 therapy post most recent alloHCT * Absolute neutrophil count (ANC) \>= 1,000/mm\^3 (without myeloid growth factors within 1 week of study entry) (performed within 28 days prior to enrollment) * Platelets \>= 50,000/mm\^3 (performed within 28 days prior to enrollment) NOTE: Platelet transfusions are not permitted within 14 days of platelet assessment * Total bilirubin =\< 2 mg/dL (exception permitted in patients with Gilbert's syndrome), unless hepatic dysfunction is a manifestation of presumed cGVHD) (performed within 28 days prior to enrollment) NOTE: Abnormal liver function tests (LFTs) (liver function panel) in the context of active cGVHD involving other organ systems may also be permitted if the treating physician documents the abnormal LFTs as being consistent with hepatic cGVHD and a liver biopsy will not be mandated in this situation * Aspartate aminotransferase (AST) =\< 3.0 x upper limit of normal (ULN) (performed within 28 days prior to enrollment) * Alanine aminotransferase (ALT) =\< 3.0 x ULN (performed within 28 days prior to enrollment) * Creatinine clearance of \>= 30 mL/min per 24-hour urine test or the Cockcroft-Gault formula (performed within 28 days prior to enrollment) * Seronegative for human immunodeficiency virus (HIV) antigen/antibody (Ag/Ab) combo, hepatitis C virus (HCV)\*, active hepatitis B virus (HBV) (surface antigen negative), and syphilis (rapid plasma reagin \[RPR\]) (performed within 28 days prior to enrollment) * If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed OR * If seropositive for HIV, HCV or HBV, nucleic acid quantitation must be performed. Viral load must be undetectable * Subjects must have negative QuantiFERON-tuberculosis (TB) Gold (QFTG) test (performed within 28 days prior to enrollment). Patients with positive QFTG test need clearance from infectious disease (ID) before enrollment * Negative for coronavirus disease 2019 (COVID-19) within 72 hours of day 0 of protocol therapy (performed within 28 days prior to enrollment) * Meets other institutional and federal requirements for infectious disease titer requirements Note Infectious disease testing to be performed within 28 days prior to start of protocol therapy * Women of childbearing potential (WOCBP): negative urine or serum pregnancy test (performed within 28 days prior to enrollment) Note: If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Cardiac function (12 lead-electrocardiography \[ECG\]): corrected QT interval (QTc) must be =\< 480 msec (performed within 28 days prior to enrollment) * Left ventricular ejection fraction \> 40% (performed within 28 days prior to enrollment) * Oxygen saturation 92% or above at room air or carbon monoxide diffusing capability test (DLCO) of 40% of best predicted (performed within 28 days prior to enrollment) Note: The above criteria only applies to participants who are not experiencing lung GVHD bronchiolitis obliterans syndrome (BOS) * Agreement by females and males of childbearing potential\* to use an effective method of birth control\*\* or abstinence from sexual activity for the course of the study through at least 6 months after the last dose of protocol therapy. * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) * Effective birth control defined as hormonal and/or barrier contraception) * ALLOGENEIC DONOR CRITERIA FOR APHERESIS DONATION: * The identified donor must be the original donor whose stem cells were used for the research participant's alloSCT * Karnofsky Performance Status (KPS) \>= 70 * Age: \>= 18 years * The donor is approved and has completed the donor evaluation per institutional guidelines (as indicated in DACT 122 - Administrative Protocol for Allogeneic Hematopoietic Progenitor Cell, Apheresis \[HPC(A)\] Collections). Additionally, donor will also be screened for the following infectious diseases: * Epstein-Barr virus (EBV), * Human herpes virus 6, 7, and 8 (HHV6, HHV7, HHV8) * Parvovirus B19 Note: ID test results for EBV, HHV6, HHV7, HHV8 and Parvovirus B19 are not necessary to proceed with the apheresis procedure but do have to be resulted and negative before participant CAR Treg infusion Exclusion Criteria: * PATIENT (RECIPIENT) EXCLUSION CRITERIA: * Immunosuppressive therapy within 28 days prior to enrollment (exception: corticosteroid) * Concurrent other investigational agents, including biologics * Vaccines within 28 days prior to enrollment * Donor lymphocyte infusion within 100 days of enrollment * No known contraindications to steroids or tocilizumab * No current active malignancy\* (Exception: Basal or squamous cell carcinoma) * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agent * Active uncontrolled infection requiring systemic antibiotics and/or anti-virals * Other autoimmune/inflammatory disorders * Clinically significant uncontrolled illness * History of vascular disease (e.g., deep vein thrombosis, stroke) * Unstable cardiac disease as defined by one of the following: * Cardiac events such as myocardial infarction (MI) within the past 6 months * NYHA (New York Heart Association) heart failure class III-IV * Uncontrolled atrial fibrillation or hypertension * Females only: Pregnant or breastfeeding * Inability to comply with protocol therapy and follow up visits * 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)