Let $G$ be a finite group and let $M$ and $N$ be two normal subgroups of $G$. Let $\rm{Aut}^{M}_{N}(G)$ 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 $\rm{Aut}^{\gamma_{n+1}(G)}_{Z_{n} (G)} (G)$ to $\rm{Aut}^{\gamma_{ n+1}(H)}_{Z_{n} (H)} (H)$, 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 $G$ of class $(n + 1)$, if $\gamma_{n+1}(G)$ is cyclic, then we prove that $\rm{Aut}^{\gamma_{n+1}(G)}_{Z_{n} (G)} (G)$ is isomorphic to the group of inner automorphisms of a quotient group of $G$.