Let G be a group. An automorphism α of G is called a commuting automorphism if xxα=xα x for all x G. We denote the set of all commuting automorphisms of G by A(G). Moreover a group G is called an AC-group if the centralizer of every non-central element of G is abelian. In this paper we show thatA(G) is a subgroup of the automorphism group of G for all finite AC-groups, p-groups of maximal class, and metacyclic p-groups.
