On equality of certain automorphism groups

Let G=H�A be a finite group, where H is a purely non-abelian subgroup of G and A is a non-trivial abelian factor of G. Then, for n?2, we show that there exists an isomorphism ?:Aut?n(G)Z(G)(G)?Aut?n(H)Z(H)(H) such that ?(Autn?1c(G))=Autn?1c(H). We also give some necessary and sufficient conditions on a finite p-group G such that Autcent(G)=Autn?1c(G) . Furthermore, for a finite non-abelian p-group G, we give some necessary and sufficient conditions for Aut?n(G)Z(G)(G) to be equal to AutZ(G)?2(G)(G).