Some properties on -automorphisms of groups

Let G be a group and IA(G) denote the group of all automorphisms of G, which induce identity map on the abelianized group Gab=G/G′. Also the group of those IA-automorphisms which fix the centre element-wise is denoted by IAZ(G). In the present article, among other results and under some condition we prove that the derived subgroups of finite p-groups, for which IAZ-automorphisms are the same as central automorphisms, are either cyclic or elementary abelian.