  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.
