Equivalence of definitions of conjugacy functor whose normalizer generates whole group with p'-core

Statement
Suppose $$G$$ is a group, $$p$$ is a prime number, and $$W$$ is a conjugacy functor for $$G$$. The following conditions are equivalent, where $$P$$ is any $$p$$-Sylow subgroup 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 $$G/O_{p'}(G)$$.

Facts used

 * 1) uses::Normality satisfies inverse image condition
 * 2) uses::Frattini's argument

(2) implies (1)
Given: A prime $$p$$, a finite group $$G$$ such that if $$K = G/O_{p'}(G)$$, then for every $$p$$-Sylow subgroup $$P$$ of $$G$$, the image of $$W(P)$$ is normal in $$K$$.

To prove: For one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$, $$\! G = O_{p'}(G)N_G(W(P))$$.

Proof: