Constructing an Automorphism From an Anti-Automorphism

We consider the following problem: Let G be a group with distinct automorphisms β and σ and an anti-automorphism α such that What can be said about G? If σ = α, σ is both an automorphism and an anti-automorphism so that G is abelian. Hence we assume that σ ≠ α. In this case, we show that G is non-abelian, but has an abelian subgroup of index 2. Conversely, for such a group G there always exist distinct automorphisms β and σ and an anti-automorphism α such that (1) holds.