Kate Burton is set to headline a Broadway revival of “The Constant Wife.”Roundabout presents the W. Somerset Maugham comedy at its American Airlines Theater. It opens June 16, with previews beginning May 27. The company’s long-extended “12 Angry Men” continues at the West 42nd Street theater through May 8. Mark Brokaw directs the Maugham play, set in 1920s London. When a wife discovers her husband is having an affair with her best friend, she turns the situation to her advantage. The first Broadway production, in 1926, starred Ethel Barrymore. Subsequent productions headlined Leopoldine Konstantin (1946), Katharine Cornell (1951) and Ingrid Bergman (1975).