Inclusion Criteria

          -  Age ≥ 18 years, ECOG performance status ≤ 1

          -  Histologically or cytologically-confirmed diagnosis of recurrent or metastatic head
             and neck squamous cell carcinoma of the oral cavity, oropharynx, hypopharynx, or
             larynx that is considered incurable by local therapies

          -  Eligible for pembrolizumab monotherapy based on the current prescribing information
             for pembrolizumab (Keytruda 2019)

          -  Must have had 0 to 2 prior therapies for R/M HNSCC.

          -  At least 1 measurable lesion (which has not been previously irradiated) according to
             RECIST v 1.1

          -  For stage 1 only: Must have at least 1 other lesion that can be biopsied and willing
             to undergo a pretreatment and on-treatment biopsy of the available tumor lesion

          -  For stage 1 only: Must be willing to undergo a pretreatment and on-treatment biopsy of
             the available tumor lesion

          -  Adequate organ function defined by the following laboratory parameters:

          -  Absolute neutrophil count (ANC) ≥ 1.5 × 109/L

          -  Platelet count ≥ 100 × 109/L

          -  Hemoglobin level ≥ 9.0 g/dL

          -  A serum creatinine level < 1.5 mg/dL, or

          -  Estimated creatinine clearance value ≥ 60 mL/min (as determined by the Cockcroft-Gault
             method) for subjects with creatinine levels > 1.5 × institutional upper limit of
             normal (ULN)

          -  Total bilirubin level ≤ 1.5 × ULN (exception: subjects with Gilbert's Syndrome may
             have a bilirubin level > 1.5 × ULN)

          -  Aspartate aminotransaminase (AST)/serum glutamic-oxaloacetic transaminase (SGOT) and
             alanine aminotransferase (ALT)/serum pyruvic transaminase (SGPT) levels ≤ 2.5 × ULN or
             ≤ 5 × ULN in subjects with liver metastases

          -  International normalized ratio (INR) or prothrombin time (PT) and activated partial
             thromboplastin time (aPTT) ≤ 1.5 × ULN, unless subject is receiving anticoagulant
             therapy in which case PT or aPTT must be within therapeutic range of intended use of
             anticoagulants

        Exclusion Criteria

          -  Previously treated with 3 or more systemic regimens given for recurrent and/or
             metastatic disease

          -  Received anticancer treatment, major surgery, or any investigational drug within 30
             days or 5 half-lives, whichever is shorter, before the start of study intervention

          -  Received radiation therapy within 14 days before the start of study intervention,
             including, in addition (if necessary), the timeframe for resolution of any actual or
             anticipated toxicities from such radiation; Palliative radiation is allowed if > 7
             days and any toxicity is ≤ Grade 1

          -  Previous treatment with a PI3K, PD-1 or PD-L1 inhibitor

          -  Have received organ or allogenic bone marrow or peripheral blood stem cell transplant

          -  History of drug-induced colitis or drug-induced pneumonitis; history or concurrent
             condition of interstitial lung disease of any severity and/or severely impaired lung
             function; tuberculosis treatment within 2 years prior to the start of study
             intervention; chronic liver disease or veno-occlusive disease/sinusoidal obstruction
             syndrome

          -  Active cytomegalovirus (CMV) or Epstein-Barr virus (EBV) infection; history of or
             known human immunodeficiency virus (HIV) infection

          -  Ongoing treatment with chronic immunosuppressants or systemic steroids or treatment
             for systemic bacterial, fungal, or viral infection

          -  Unable to receive prophylactic treatment for pneumocystis, herpes simplex virus (HSV),
             or herpes zoster (VZV) at screening

          -  Concurrent administration of medications or foods that are strong inhibitors or
             inducers of cytochrome P450 3A (CYP3A). No prior use within 2 weeks before the start
             of study intervention Received a live or live attenuated vaccine within 6 weeks of
             first dose of duvelisib

          -  Unable to receive prophylactic treatment for pneumocystis, herpes simplex virus (HSV),
             or herpes zoster (VZV) at screening

          -  Any active gastrointestinal dysfunction interfering with the subject's ability to be
             administered oral medications

          -  Known active central nervous system metastases and/or carcinomatous meningitis

          -  QT interval > 500 ms (except for subjects with a right or left bundle branch block)

          -  New York Heart Association Class III or IV congestive heart failure