Stabiliser of a Subset of Center I have no clue for the following problem: 

Let $G$ be a finite group, $p$ a prime number, $S$ a Sylow $p$ subgroup of $G$. Let $N$ be the normalizer of $S$ inside $G$. Let $X, Y$ two subsets of $Z(S)$ (center of $S$) such that $\exists g \in G, gXg^{-1}= Y$. Then we need to show that $\exists n \in N$ such that $gxg^{-1} = nxn^{-1}, \forall x \in X$. 

So I guess first I can assume $X, Y$ to be subgroups by taking the smallest subgroup containing them. Then I have no clue. 

Because $X$ is contained in $Z(P)$, it follows that $N_G(X)$ contains $P$. That means that $N_G(Y) = N_G({}^gX)$ must contain ${}^gP$. But it also contains $P$, since $Y$ is central in $P$.
Now, notice that both $P$ and ${}^gP$ are Sylow $p$-subgroups of $N_G(Y)$. Can you take it from there?
