I've had a lot of trouble with this question, as well as my advisor.
Let x and y be two elements of $Z(P)$, where $P$ is a Sylow $p$-subgroup of $G$. If $x$ and $y$ are conjugate in $G$, prove that $x$ is conjugate to $y$ in $N_G(P)$, the normalizer of $P$ in $G$.