Characteristic p-functor that controls normal p-complements

Definition
Suppose $$p$$ is a prime number and $$W$$ is a characteristic p-functor. We say that $$W$$ controls normal $$p$$-complements in a finite group $$G$$ if the following holds: if there exists a $$p$$-Sylow subgroup $$P$$, such that $$N_G(W(P))$$ possesses a defining ingredient::normal p-complement, $$G$$ also possesses a normal $$p$$-complement.

We say that $$W$$ controls normal $$p$$-complements in general if it controls normal $$p$$-complements in every finite group.

Facts

 * Characterization of minimal counterexamples to a characteristic p-functor controlling normal p-complements
 * The generalized Glauberman-Thompson normal p-complement theorem gives sufficient conditions for a characteristic p-functor to control normal p-complements in every finite group.
 * The Glauberman-Thompson normal p-complement theorem states that the ZJ-functor controls normal $$p$$-complements for odd primes $$p$$.
 * The Glauberman-Solomon normal p-complement theorem states that the D*-subgroup functor controls normal $$p$$-complements for odd primes $$p$$.