 This yields a map PGL(2, 5) → S6 as a transitive subgroup.Identifying PGL(2, 5) with S5 and the projective special linear group PSL(2, 5) with A5 yields the desired exotic maps S5 → S6 and A5 → A6.Following the same philosophy, one can realize the outer automorphism as the following two inequivalent actions of S6 on a set with six elements