We provide existence criteria and characterizations for outer inverses in an involutive semigroup belonging to the prescribed Green’s \({\mathscr {R}}\) -, \({\mathscr {L}}\) - and \({\mathscr {H}}\) -classes. We show that the existence criteria for such outer inverses are simultaneously the existence criteria for various types of generalized inverses, such as \(\{1,4\}\) -inverses, \(\{1,3\}\) -inverses, Moore–Penrose inverses, core inverses and dual core inverses. We also provide several new characterizations of the mentioned types of generalized inverses, as well as characterizations of \(\{2,4\}\) -inverses and \(\{2,3\}\) -inverses.
