Equality of certain automorphism groups of finite $p$-groups

Let $G$ be a finite $p$-group and let Aut$(G)$ denote the full automorphism group of $G$. In the recent past, there has been interest in finding necessary and sufficient conditions on $G$ such that certain subgroups of Aut$(G)$ are equal. We prove a technical lemma and, as a consequence, obtain some new results and short and alternate proofs of some known results of this type.