On the Equality of Certain Subgroups of the Automorphism Groups of Finite $p$-Groups

Let $G$ be a finite non-Abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called an IA-automorphism if $x^{-1}\alpha(x)\in G^{\prime} $ for all $x\in G$. An automorphism $\alpha$ of $G$ is called an absolute central automorphism if, for all $x\in G$, $x^{-1}\alpha(x)\in L(G)$, where $L(G)$ is the absolute center of $G$. Let $C_{\text{IA}(G)}(Z(G))$ and $C_{\text{Var}(G)}(Z(G))$ denote, respectively, the group of all IA-automorphisms and the group of all absolute central automorphisms of $G$ fixing the center $Z(G)$ of $G$ elementwise. We give necessary and sufficient conditions on a finite $p$-group $G$ under which $C_{\text{IA}(G)}(Z(G))$ = $C_{\text{Var}(G)}(Z(G))$.