Conjugacy functor whose normalizer generates whole group with p'-core controls fusion

Statement
Suppose $$G$$ is a finite group, $$p$$ is a prime number, and $$W$$ is a $$p$$- conjugacy functor on $$G$$ whose normalizer, along with the p'-core, generates the whole group. Explicitly, this means that if $$P$$ is a $$p$$-Sylow subgroup of $$G$$:

$$\! G = O_{p'}(G)N_G(W(P))$$.

Then, $$W$$ controls fusion in $$G$$. In other words, any two subsets of $$P$$ that are conjugate in $$G$$ are also conjugate in $$N_G(W(P))$$.