On commuting automorphisms of finite p-groups of co-class 3, when p>2

Let G be a group. An automorphism of G is called a commuting automorphism if (g), g=1 for all g G. Let A (G) denote the set of all commuting automorphisms of G. A group G is said to be an A (G) -group if A (G) forms a subgroup of Aut (G), where Aut (G) denotes the group of all automorphisms of G. In Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 5, 57-60 Rai proved that a finite p-group G of co-class 2 for an odd prime p is an A (G) -group. We prove that a finite p-group G of co-class 3 for an odd prime p, under some conditions, is an A (G) -group.