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

Statement
Suppose $$G$$ is a group, $$p$$ is a prime number, and $$W$$ is a characteristic p-functor. The following are equivalent:


 * 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 $$K$$.

Facts used

 * 1) uses::Equivalence of definitions of conjugacy functor whose normalizer generates whole group with p'-core
 * 2) uses::Equivalence of normality and characteristicity conditions for conjugacy functor

Equivalence of (1) and (2)
This is direct from Fact (1). Note that that only uses the conjugacy functor.

Equivalence of (2) and (3)
This requires the observation that for the quotient map $$G \to G/O_{p'}(G)$$, Sylow subgroups of $$G$$ map isomorphically to Sylow subgroups of $$G/O_{p'}(G)$$, so the quotient map commutes with $$W$$.

Equivalence of (3) and (4)
This is direct from Fact (2).