Subsets of a well-placed tame intersection that are conjugate in its normalizer are conjugate in the normalizer of the conjugacy functor

Statement
Suppose $$G$$ is a finite group, $$P,Q$$ are $$p$$-Sylow subgroups of $$G$$ such that $$P \cap Q$$ is a fact about::well-placed tame Sylow intersection: it is a fact about::tame Sylow intersection that is well-placed in $$P$$. Suppose $$W$$ is a fact about::conjugacy functor on the $$p$$-subgroups of $$G$$ such that $$W$$ controls fusion in the normalizer of every non-identity $$p$$-subgroup of $$G$$. Then, if $$A,B$$ are subsets of $$P \cap Q$$ that are conjugate in $$N_G(P \cap Q)$$, $$A,B$$ are also conjugate in $$N_G(W(P))$$.

Facts used

 * 1) uses::Nilpotent implies normalizer condition
 * 2) uses::Conjugacy functor gives normalizer-relatively normal subgroup
 * 3) uses::Normality satisfies intermediate subgroup condition
 * 4) uses::Characteristic of normal implies normal

Applications

 * Control of fusion is local: This is a slightly stronger result that follows by combining the given result with Alperin's fusion theorem in terms of well-placed tame intersections.

Proof
Given: A finite group $$G$$, a prime $$p$$. A conjugacy functor $$W$$ on the $$p$$-subgroups of $$G$$. $$P,Q$$ are $$p$$-Sylow subgroups such that $$P \cap Q$$ is a well-placed tame intersection. $$W$$ controls fusion in the normalizer of every non-identity $$p$$-subgroup of $$G$$. $$A,B$$ are subsets of $$P \cap Q$$ that are conjugate in $$N_G(P \cap Q)$$.

To prove: $$A,B$$ are conjugate in $$N_G(W(P))$$.

Proof: