Conjugacy functor that controls fusion

Definition
Suppose $$G$$ is a finite group and $$p$$ is a prime number. Suppose $$W$$ is a conjugacy functor on the nontrivial $$p$$-subgroups of $$G$$. We say that $$W$$ controls $$p$$-fusion in $$G$$ if, for any $$p$$-Sylow subgroup $$P$$ of $$G$$, $$P$$ is a weak subset-conjugacy-determined subgroup inside $$N_G(W(P))$$.

(Note that $$P$$ is contained in $$N_G(W(P))$$ because $$W(P)$$ is normal in $$P$$ by the conjugation-invariance property that conjugacy functors have to satisfy. In fact, $$N_G(P) \le N_G(W(P))$$ by the fact that conjugacy functor gives normalizer-relatively normal subgroup).

Related group properties

 * Group in which the ZJ-functor controls fusion

Facts

 * Control of fusion is local: If $$W$$ is a conjugacy functor such that the restriction of $$W$$ to the normalizer of any non-identity $$p$$subgroup controls fusion in that subgroup, then $$W$$ controls fusion in the whole group.