On sub-central automorphisms of finite p-groups

Let G be a group. For a normal subgroup X of G, let AutX(G) denote the subgroup of Aut(G) centralizing G/X. In this paper, we give necessary and sufficient conditions for a finite purely non-abelian p-group G to satisfy AutH(G)=Autγn+1(G)(G), where 1≠H≤Z(G), and consequently derive the main result of Shabani-Attar [Citation15]. As a consequence of our results, we also characterize all finite non-abelian p-groups G of order up to p6 such that AutL(G)(G)=Autγ2(G)(G), where L(G) denotes the absolute center of G.