A study on Ivar(G) of a p-group fixing certain subgroups

‎Let $G$ be a group, and $M$ and $N$ be two normal subgroups of $G$. In this paper, we first introduce a subgroup $\mathcal{E}(G)$ and consider the set of all automorphisms of $G$ which centralize $G/M$ and $N$. Then we investigate the conditions in which this set of automorphisms with different $M$ and $N$ is equal or to be equal with $Ivar(G)$.