Group in which the ZJ-functor controls fusion for a prime

Definition
Suppose $$G$$ is a finite group and $$p$$ is a prime number. We say that $$G$$ is a group in which the ZJ-functor controls fusion for $$p$$ if the defining ingredient::ZJ-functor, viewed as a conjugacy functor on $$G$$, controls fusion in $$G$$ with respect to $$p$$. In other words, given a $$p$$-Sylow subgroup $$P$$ of $$G$$ and two subsets $$A,B$$ of $$P$$ that are conjugate in $$G$$, the subsets $$A,B$$ are conjugate in $$N_G(Z(J(P))$$.

Stronger properties

 * Weaker than::Group of Glauberman type for a prime
 * Weaker than::Group in which every p-local subgroup is of Glauberman type