Conjugacy functor gives normalizer-relatively normal subgroup

Definition
Suppose $$G$$ is a finite group, $$p$$ is a prime number, $$W$$ is a conjugacy functor for the prime $$p$$ in $$G$$, and $$P$$ is a nontrivial $$p$$-subgroup of $$G$$ (not necessarily a Sylow subgroup, though we typically apply this to Sylow subgroups). Then, $$W(P)$$ is a normal subgroup of $$N_G(P)$$.

Proof
Given: Finite group $$G$$, prime $$p$$, $$p$$-conjugacy functor $$W$$, nontrivial $$p$$-subgroup $$P$$. $$u \in N_G(P)$$.

To prove: $$uW(P)u^{-1} = W(P)$$.

Proof: