On $n^{th}$ class preserving automorphisms of $n$-isoclinism family

Abstract: Let $G$ be a finite group and $M,N$ be two normal subgroups of $G$. Let $Aut_N^M(G)$ denote the group of all automorphisms of $G$ which fix $N$ element wise and act trivially on $G/M$. Let $n$ be a positive integer. In this article we have shown that if $G$ and $H$ are two $n$-isoclinic groups, then there exists an isomorphism from $Aut_{Z_n(G)}^{\gamma_{n+1}(G)}(G)$ to $Aut_{Z_n(H)}^{\gamma_{n+1}(H)}(H)$, which maps the group of $n^{th}$ class preserving automorphisms of $G$ to the group of $n^{th}$ class preserving automorphisms of $H$. Also, for a nilpotent group of class at most $(n+1)$, with some suitable conditions on $\gamma_{n+1}(G)$, we prove that $Aut_{Z_n(G)}^{\gamma_{n+1}(G)}(G)$ is isomorphic to the group of inner automorphisms of a quotient group of $G$.