Some aspects of marginal automorphisms of a finite $p$-group

Let $F$ be a free group, $mathcal{V}$ be a variety of groups defined by the set of laws $Vsubseteq F$ and $G$ be a finite $mathcal{V}$-nilpotent $p$-group. The automorphism $alpha$ of $G$ is said to be a marginal automorphism (with respect to $V$), if for all $xin G$, $x^{-1}x^{alpha}in V^{star}(G)$, where $V^{star}(G)$ denotes the marginal subgroup of $G$. An automorphism $alpha$ of $G$ is called an IA-automorphism if $x^{-1}x^{alpha}in G'$ for each $xin G$. An automorphism $alpha$ of $G$ is called a class preserving if for all $xin G$, there exists an element $g_xin G$ such that $x^{alpha}=g_x^{-1}xg_x$. Let $operatorname{Aut}^{V^{star}}(G)$, $operatorname{Aut}^{G'}(G)$ and $operatorname{Aut}_c(G)$ respectively, denote the group of all marginal automorphisms, IA-automorphisms and class preserving automorphisms of $G$. In this paper, first we give a necessary and sufficient condition on a finite $mathcal{V}$-nilpotent $p$-group $G$ such that each marginal automorphism of $G$ fixes the center of $G$ element-wise. Then we characterize all finite $mathcal{V}$-nilpotent $p$-groups $G$ such that $operatorname{Aut}^{V^{star}}(G)=operatorname{Aut}^{G'}(G)$. Finally, we obtain a necessary and sufficient condition for a finite $mathcal{V}$-nilpotent $p$-group $G$ such that $operatorname{Aut}^{V^{star}}(G)=operatorname{Aut}_c(G)$.