I'll state the result I'm trying to prove, progress I've made, and the two questions I have which will help me solve it. The question is originally motivated by studying defect groups in modular representation theory.
Let $D$ be be a subgroup of a finite group $G$ with the following property:
If $S$ is a Sylow $p$-subgroup of $G$ containing $D$ then there is $c\in C(D)$ such that $D=S\cap cSc^{-1}$.
Prove that $D$ is the largest normal $p$-subgroup of $N_G(D)$.
The start of my attempt at a proof: It will suffice to show that $D$ is the intersection of Sylow $p$-subgroups of $N_G(D)$, since every normal $p$-subgroup of $N_G(D)$ is contained in every Sylow $p$-subgroup of $N_G(D)$, by Sylow II. We do not however know that $cSc^{-1}$ is Sylow (Sylow II states that all Sylows are conjugate but not necessarily that all conjugates are Sylow; right?)
Questions:
Q1: If we let $T$ be a Sylow $p$-subgroup of $N_G(D)$, can we necessarily find a Sylow $p$-subgroup $S$ of $G$ containing $T$?
Q2: If $T$ is a Sylow $p$-subgroup of $N_G(D)$, must $T\supseteq D$ and $cTc^{-1}\supseteq D$ for any $c\in C(D)$?
(If the answer to these two is yes then the result will follow).