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 ϕ:Autγn(G)Z(G)(G)→Autγn(H)Z(H)(H) such that ϕ(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 Autγn(G)Z(G)(G) to be equal to AutZ(G)γ2(G)(G).by Surjeet Kour and Vishakh