Characteristic p-functor whose normalizer generates whole group with p'-core

Definition
Suppose $$G$$ is a group, $$p$$ is a prime number, and $$W$$ is a characteristic p-functor. We say that $$W$$ is a characteristic p-functor whose normalizer generates whole group with p'-core for $$G$$ if the corresponding conjugacy functor on $$G$$ is a defining ingredient::conjugacy functor whose normalizer generates whole group with p'-core. Equivalently, $$W$$ satisfies the following equivalent conditions for one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$:


 * 1) For one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$, $$O_{p'}(G)N_G(W(P)) = G$$
 * 2) For one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$, the image of $$W(P)$$ in the quotient $$G/O_{p'}(G)$$ is a normal subgroup of $$G/O_{p'}(G)$$.
 * 3) For one (and hence every) $$p$$-Sylow subgroup $$Q$$ of $$K = G/O_{p'}(G)$$, $$W(Q)$$ is a normal subgroup of $$K$$.
 * 4) For one (and hence every) $$p$$-Sylow subgroup $$Q$$ of $$K = G/O_{p'}(G)$$, $$W(Q)$$ is a characteristic subgroup of $$K$$.