Laboratory-Treated (Central Memory/Naive) CD8+ T Cells in Treating Patients With Newly Diagnosed or Relapsed Acute Myeloid Leukemia

Inclusion Criteria: * Patients with (non-M3) acute myeloid leukemia (AML) * Patients must be \>= 15 kg * Patients or parents/legal guardian must be able to give informed consent * Patients must be able to provide blood and marrow samples and to undergo the procedures required for this protocol * Elevated expression of WT1 in pre-treatment bone marrow or peripheral blood by either of two methods: * Increased expression of WT1 determined if the number of copies of WT1 divided by the number of copies of ABL x 10\^4 is \> 250 for bone marrow, or \> 50 for peripheral blood; * Demonstration of WT1 overexpression will be determined by immunohistochemical staining (IHC) in blasts as compared to adjacent normal myeloid and erythroid precursors, as determined by a Fred Hutchinson/Seattle Cancer Care Alliance pathologist * Demonstration of disease response to induction chemotherapy, in that patients must have achieve a morphologic remission (marrow that is at least 10% cellular with \< 5% blasts on morphologic review) after 1-2 induction cycles, regardless of minimal residual disease or incomplete hematologic recovery (CRi)/incomplete platelet recovery (CRp) status * Determination of "high-risk" disease; subjects must meet one of the determinants of "high-risk disease", in terms of being at very high risk for relapse without allogeneic stem cell transplant, as per one of the follow criteria: * A designation of "adverse" risk disease at the time of diagnosis, as defined by cytogenetic and molecular abnormalities specifically outlined in the 2017 European LeukemiaNet (ELN) guidelines for diagnosis and management of AML; these patients will meet "high-risk" designation, regardless of minimal residual disease or CRi/CRp status * Relapsed leukemia; patients with cytogenetic or molecular classification other that adverse risk by ELN who go on to demonstrate disease relapse after a minimum duration of remission of 6 months, but who then attain a second complete remission with repeat induction chemotherapy; these patients will meet "high-risk" designation, regardless of minimal residual disease or CRi/CRp status * Minimal residual disease, as defined by having detectable disease by one of the following criteria, but otherwise being in morphologic remission * MRD by flow cytometry at any time after induction chemotherapy or during consolidation chemotherapy, when patients are otherwise classified as being in morphologic remission, and as defined by any abnormal myeloid blasts identified by flow cytometric analysis * Cytogenetic MRD, as defined by a disease-specific abnormal karyotype at any point in patients who are otherwise in morphologic remission * Molecular minimal residual disease (MRD) with one of the following markers, as specified below, in patients who are otherwise in morphologic remission: * A normalized copy number (NCN) of \> 0.001 for CBFB/MYHI1 or a normalized copy number (NCN) of \> 0.050 for AMLI/ETO (RUNXI/RUNX1T1) after at least 4 cycles of consolidation chemotherapy * A \< 2 log reduction in either CBFB/MYH11 or AMLI/ETO (RUNX1/RUNX1T1) in the bone marrow at the time of post-induction disease restaging (immediately after 1-2 cycles of induction therapy) * CRi/CRp, as defined by neutrophil count \< 1000/ul (CRi) and/or platelet count \< 100,000/ul (CRp), but otherwise being in morphologic remission; in pediatric patients, a platelet threshold of \< 80,000/ ul will be used, as per consensus pediatric response criteria * Human leukocyte antigen (HLA)-A\*02:01 expression must be present for patient to be on treatment arm, HLA-A\*02:01 expression absent in patients designated to observation arm ELIGIBILITY FOR APHERESIS/BLOOD COLLECTION: • HLA-A\*02:01 expression ELIGIBILITY FOR TREATMENT WITH TCRC4-TRANSDUCED CD8+ CELLS * Response to therapy and completion of at least one cycle of consolidation therapy, and with disease status meeting one of the aforementioned "high-risk" criteria at the time of post-induction disease restaging as already outlined above * Hematologic recovery from induction and other post-remission therapy (absolute neutrophil count \[ANC\] \> 200/ul, platelet count \> 20,000/ul) at the time of the study intervention * No plan for allogeneic stem cell transplantation within 3 months * Elevated expression above baseline of WT1 in bone marrow or peripheral blood * Additionally, patients treated in stage 1, cohort #3 must be Epstein-Barr virus (EBV) seropositive, given the inclusion of T cells derived from an EBV-specific subset in this group * Continued morphologic remission (\< 5% blasts in the marrow, no circulating blasts or known extramedullary relapse) within 6 weeks of receiving the study intervention (specified as T cell infusion for cohort 1, or the start of lymphodepleting chemotherapy for cohorts 2 and 3) * ELIGIBILITY FOR OBSERVATION ARM * The patient meets all of the eligibility criteria for enrollment, but lacks expression of HLA-A\*0201 as is needed to be enrolled on the treatment arm and for apheresis Exclusion Criteria: * Active autoimmune disease (e.g. systemic lupus erythematosus, vasculitis, infiltrating lung disease, inflammatory bowel disease) in which possible progression during treatment would be considered unacceptable by the investigators * Previous allogeneic hematopoietic cell transplant (HCT) * Any condition or organ toxicity deemed by the principal investigator (PI) or the attending physician to place the patient at unacceptable risk for treatment on the protocol * Pregnant women, nursing mothers, men or women of reproductive ability who are unwilling or unable to use effective contraception or abstinence; women of childbearing potential must have a negative pregnancy test within two weeks prior to enrollment and initiation of treatment * Clinically significant and ongoing immune suppression including, but not limited to: systemic immunosuppressive agents such as cyclosporine or corticosteroids (at an equivalent dose of 0.5 mg prednisone/kg per day, or higher), chronic lymphocytic leukemia (CLL), uncontrolled human immunodeficiency virus (HIV) infection (untreated or detectable viral load within 3 months of enrollment) * Acute promyelocytic leukemia (M3 leukemia, per French-American-British classification) EXCLUSION FOR TREATMENT WITH TCRC4-TRANSDUCED CD8+ CELLS * Unable to generate antigen-specific WT1-specific CD8+ T cells for infusions; however, if a lower than planned number of cells is available, the patient will have the option to receive the generated WT1-specific T cells * Systemic steroids should be stopped 2 weeks before the start of treatment; topical and inhaled steroids are allowed * Symptomatic and refractory central nervous system (CNS) leukemia * Absolute neutrophil count (ANC) \< 200/ul prior to treatment * Platelets \< 20,000/ul prior to treatment * If a patient meets other treatment eligibility but otherwise demonstrates delayed or poor recovery of peripheral blood counts to the above neutrophil and/or platelet thresholds, then treatment with the T cell intervention will be allowed if: * Neutrophil and/or platelet counts remain below the thresholds after a period of at least 6 weeks from last systemic chemotherapy; OR neutrophil and/or platelet counts remain below the thresholds in the setting of a maintenance therapy, such as midostaurin; and * The patient has detectable leukemia (e.g. flow cytometry positive or MRD by FISH or molecular testing); and * The P.l. or treating physician documents that the likely cause of cytopenias is underlying disease as opposed to another cause (e.g. medication) * Ongoing \>= grade 3 cardiac, pulmonary, renal, gastrointestinal or hepatic toxicities according to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) version 4 toxicity criteria * Karnofsky performance status score (age \>= 16 years) or Lansky play score (age \< 16 years) =\< 40% * Medical or psychological conditions that, according to the PI, would make the patient unsuitable candidate for cell therapy * Pregnancy or breast-feeding; women of childbearing potential must have a negative serum or urine beta-human chorionic gonadotropin (hCG) pregnancy test result within 14 days before the first dose of WT1-specific T cell infusions; woman of non-childbearing potential will be defined as being postmenopausal greater than one year or who have had a bilateral tubal ligation or hysterectomy; all recipients of WT1-specific T cells will be counseled to use effective birth control during participation in this study and for 12 months after the last T cell infusion * Treatment with alemtuzumab or other T cell-depleting antibodies within 6 months of T cell therapy * Documented new infection within 24 hours of T cell infusion, or concern for new infection as suggested by an oral temperature \> 38.2 degrees Celsius (C) within 24 hours of T cell infusion * ln patients who have T cells delayed because of development of fever (oral temperature \> 38.2 degrees C) and who subsequently become afebrile (38.2 degrees C or less) for 24 hours without a documented infection, T cells may be administered * ln patients with a documented new infection within 24 hours of planned T cell infusion, they may go on to receive T cells after administration of directed antibiotic therapy and if they subsequently remain afebrile (38.2 degrees C or less) for at least 24 hours, and if it is deemed clinically appropriate by the Pl * Pre-existing infections requiring chronic maintenance therapy (e.g. chronic hepatitis B virus \[HBV\] or treated bacterial infections) are not an exclusion for T cell infusion as long as patients are on appropriate antimicrobial therapy for at least 1 month (e.g. for chronic hepatitis B or C viral infection) and who remain afebrile and without symptomatic evidence for uncontrolled chronic infection within 24 hours of T cell infusion; patients should also have a negative HIV test by viral load within 3 months of treatment