This is exercise 5.6 from Algebra, Isaacs.
Let $G$ be a finite group with $|G|>1$, and suppose $P\subseteq \operatorname{Aut}(G)\ $ is a $p$-subgroup. Show that there exists some nontrivial Sylow $q$-subgroup $Q$ of $G$ (for some prime $q$) such that $\sigma(Q)=Q$ for all $\sigma \in P \ $.
I can solve the case $ p \mid |G| \ $, but not the other.