On equality of certain automorphism groups

Let G = H×A be a 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 (Formula presented.) such that (Formula presented.). Also, for a finite non-Abelian p-group G satisfying a certain natural hypothesis, we give some necessary and sufficient conditions for (Formula presented.). Furthermore, for a finite non-Abelian p-group G, we study the equality of Autcent(G) with (Formula presented.).