Let $G$ be a finite group and $p$ be a prime. Let $H$ be a subgroup of $G$ which contains $N_G(P)$ for some Sylow $p$-subgroup $P$ of $G$. Suppose $P \subseteq H^g$ for some $g \in G$. Prove that $g \in H$.
With the assumptions I can prove that $N_G(H)=H$. What should I do then?
Thanks in advanced.