  Assume $G$ is a finite $p$-group, and let $S$ be a Sylow $p$-subgroup of
$\operatorname{Aut}(G)$ with $\exp(S)=q$. We prove that if $G$ is of class $c$,
then $\exp(G)|p^{\ceil{\log_pc}}q^3$, and if $G$ is a metabelian $p$-group of
class at most $2p-1$, then $\exp(G)|pq^3$.
