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

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


 * 1) $$O_{p'}(G)N_G(W(P)) = G$$
 * 2) The image of $$W(P)$$ in the quotient $$G/O_{p'}(G)$$ is a normal subgroup of $$G/O_{p'}(G)$$.

Related notions

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