Conjugacy functor that controls strong fusion

Definition
Suppose $$G$$ is a finite group and $$p$$ is a prime number. Suppose, further, that $$W$$ is a defining ingredient::conjugacy functor on $$G$$. We say that $$W$$ controls strong fusion on $$G$$ if, for any $$p$$-Sylow subgroup $$P$$ of $$G$$, $$P$$ is a defining ingredient::subset-conjugacy-determined subgroup inside $$N_G(W(P))$$. In other words, given two subsets $$A$$ and $$B$$ in $$P$$ that are conjugate by $$g \in G$$, there exists $$h \in N_G(W(P))$$ such that conjugation by $$h$$ has the same effect as conjugation by $$g$$ on every element of $$A$$.