Interleukin-15 (IL-5) in Combination With Avelumab (Bavencio) in Relapsed/Refractory Mature T-cell Malignancies

* INCLUSION CRITERIA: * Patients must have histologically or cytologically proven relapsed/refractory T-cell lymphoma other than adult T-cell leukemia/lymphoma (ATLL), angioimmunoblastic T-cell lymphoma (AITL), peripheral T-cell lymphoma T follicular helper phenotype (PTCL-TFH), or enteropathy-associated T-cell lymphoma (EATL), confirmed by the Laboratory of Pathology, National Cancer Institute (NCI) * Patients with CD30 (also known as TNFRSF8) + mycosis fungoides/Sezary syndrome (MF/SS) or CD30+ anaplastic large cell lymphoma (ALCL) must have relapsed after or become intolerant to treatment with brentuximab vedotin. * A formalin fixed tissue block or 15 slides of tumor sample (archival or fresh) must be available for performance of correlative studies. NOTE: Patients must be willing to have a tumor biopsy if prior tissue or adequate archival tissue is not available (i.e., post-enrollment and prior to treatment). * Disease must be measurable with at least one measurable lesion by response evaluation criteria in lymphoma (RECIL) 2017 or Modified Severity-Weighted Assessment Tool (mSWAT) criteria or have an abnormal clonal T-cell population detectable by peripheral blood flow cytometry * Age greater than or equal to 18 years NOTE: Because no dosing or adverse event data are currently available on the use of rhIL-15 in combination with avelumab in patients \<18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Eastern Cooperative Oncology Group (ECOG) performance status less than or equal to 1 * Adequate organ and marrow function as defined: * Absolute neutrophil count greater than 1,000/mcL * Absolute lymphocyte count greater than or equal to 500/mcL * Hemoglobin greater than or equal to 9 g/dL * Platelets greater than 100,000/mcL * Total bilirubin less than or equal to 1.5 X institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) Serum glutamic oxaloacetic transaminase (SGOT)/Alanine aminotransferase (ALT) Serum glutamic pyruvic transaminase (SGPT) less than or equal to 2.5 X institutional ULN * Serum creatinine less than or equal to 1.5 X institutional ULN OR * Creatinine clearance greater than or equal to 50 mL/min/1.73 m\^2 for patients with creatinine levels greater than 1.5 institutional ULN * Negative serum or urine pregnancy test at screening for women of childbearing potential (WOCBP) NOTE: WOCBP is defined as any female who has experienced menarche and who has not undergone successful surgical sterilization or who is not postmenopausal. WOCBP must have a negative pregnancy test (human chorionic gonadotropin (HCG) blood or urine) during screening. * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and 30 days after completion of rhIL-15 and avelumab administration. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. * Ability of subject to understand and the willingness to sign a written informed consent document EXCLUSION CRITERIA: * Patients with the following T-cell leukemias/lymphomas: adult T-cell leukemia/lymphoma (ATLL), angioimmunoblastic T-cell lymphoma (AITL), peripheral T-cell lymphoma T follicular helper phenotype (PTCL-TFH), and enteropathy-associated T-cell lymphoma (EATL). * Chemotherapy and anti-tumor antibodies within 4 weeks (6 weeks for nitrosoureas or mitomycin C); other tumor-directed systemic therapy and radiation therapy within 2 weeks. * Persisting toxicity related to prior therapy of grade \> 1, with the exception of the following: alopecia, sensory neuropathy grade less than or equal to 2, or other grade less than or equal to 2 not constituting a safety risk based on investigator's judgment * Patients who are receiving any other investigational agents * Patients who have had prior therapy with any antibody/drug targeting programmed cell death protein 1 (PD-1)/programmed death-ligand 1 (PD-L1) Tcell coregulatory proteins (immune checkpoints) * Current use of immunosuppressive medication, EXCEPT for the following: * Intranasal, inhaled, topical steroids, or local steroid injection (e.g., intra-articular injection) * Systemic corticosteroids at physiologic doses less than or equal to 10 mg/day of prednisone or equivalent; or, * Steroids as premedication for hypersensitivity reactions (e.g., computed tomography (CT) scan premedication) * Patients with known central nervous system (CNS) involvement should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events * Patients with previous malignant disease other than the target malignancy within the last 5 years with the exception of basal or squamous cell carcinoma of the skin or cervical carcinoma in situ * Patients with history of any organ transplantation, including allogenic stem cell transplantation * Received a live vaccine within 4 weeks of the first dose of avelumab. Vaccination with a live vaccine while on trial is prohibited. NOTE: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist) are live attenuated vaccines, and are not allowed * Patients with history of allergic reactions attributed to compounds of similar chemical or biologic composition to recombinant human Interleukin-15 (rhIL-15) or avelumab * Patients with uncontrolled intercurrent illness including, but not limited to, ongoing or active infection requiring systemic therapy, or psychiatric illness/social situations that would limit compliance with study requirements * Inability or refusal to practice effective contraception during therapy or the presence of pregnancy or active breastfeeding. Based on its mechanism of action, avelumab can cause fetal harm when administered to a pregnant woman. Animal studies have demonstrated that inhibition of the PD-1/PD-L1 pathway can lead to increased risk of immune-mediated rejection of the developing fetus resulting in fetal death. These potential risks may also apply to other agents used in this study * Patients with active bacterial infections, documented human immunodeficiency virus (HIV) infection or positive screening serology, polymerase chain reaction (PCR) evidence for active or chronic hepatitis B or hepatitis C, or positive screening hepatitis B virus (HBV)/hepatitis C virus (HCV) serology without documentation of successful curative treatment * Patients with active or history of any autoimmune disease, unrelated to their malignancy, including asthma requiring chronic inhaled or oral corticosteroids, or with history of asthma requiring mechanical ventilation; patients with a history of mild asthma that are on or can be switched to non-corticosteroid bronchodilator regimens are eligible * Cardiovascular disease: Clinically significant (i.e., active) cardiovascular disease: cerebral vascular accident/stroke (\< 6 months prior to enrollment), myocardial infarction (\< 6 months prior to enrollment), unstable angina, congestive heart failure (greater than or equal to New York Heart Association Classification Class II), or serious cardiac arrhythmia requiring medication * Other severe acute or chronic medical conditions including immune colitis, inflammatory bowel disease, immune pneumonitis, pulmonary fibrosis or psychiatric conditions including recent (within the past year) or active suicidal ideation or behavior; or laboratory abnormalities that may increase the risk associated with study participation or study treatment administration or may interfere with the interpretation of study results and, in the judgment of the investigator, would make the patient inappropriate for entry into this study