Fusion in the normaliser of a Sylow subgroup. Let $z \in P \cap Z(N_{G}(P))$ for some $P \in Syl_{p}(G)$ and $z \notin P'$. If $tz^{n}t^{-1} \in P$, for some $t \in G$, then $tz^{n}t^{-1} = z^{n}$.
I have tried a lot solving the above question, but was unable to do it. What I know is that, $N_{G}(P)$ controls the fusion of $C_{G}(P)$, so if I can show that $tz^{n}t^{-1} \in C_{G}(P)$, then the rest follows from the Burnside's lemma. But I am not able to show that $tz^{n}t^{-1} \in C_{G}(P)$.
 A: The same question was asked by the OP on mathoverflow: link.
The question is motivated by Exercise 9.4 in "Algebra: A Graduate Course" by Isaacs.  This exercise asks to prove:

Suppose that $P$ is a $p$-Sylow and $z \in Z(N_G(P)) \cap P$ and $z \notin P'$. Show that $z \not\in G'$.

Unfortunately this exercise and the hint seems to be wrong.
Equivalently, we are asked to prove that $Z(N_G(P)) \cap P \cap G' \leq P'$. This is true when $P$ is abelian, by transfer one can show that $Z(N_G(P)) \cap P \cap G' = 1$ in this case (this is corollary 9.16 in the book). But in general it seems to be false.
By a computer calculation with MAGMA or GAP, you can find that SmallGroup(96,64) is a group $G$ that works as a counterexample. We have:

*

*$|G| = 96 = 2^5 \cdot 3$

*For $2$-Sylow $P$, we have $N_G(P) = P$. Furthermore $Z(P) \leq G'$.

*Thus $Z(N_G(P)) \cap P \cap G' = Z(P)$.

*$Z(P) \not\leq P'$. (Here $Z(P)$ has order $4$, and $Z(P) \cap P'$ has order $2$.)

Thus $Z(N_G(P)) \cap P \cap G' \not\leq P'$.
Your question (and the version before it was edited) is based on the hint, which suggests the following:

Let $z \in Z(N_G(P)) \cap P$ and $z \not\in P'$. If $tz^nt^{-1} \in P$, then $tz^nt^{-1} = z^n$.

This is also false. For example, in $G$ as above, take $z \in Z(P)$ with $z \not\in P'$. Then there are three $G$-conjugates of $z$ in $P$, so the claim fails with $n = 1$.
The original question, before you edited it, was the following:

Let $z \in Z(N_G(P)) \cap P$. If $tzt^{-1} \in P$, then $tzt^{-1} = z$.

This fails with the above example. An easier example is $G = S_4$ with $P$ a Sylow $2$-subgroup.
