Outer automorphism group of finite p-group that is not elementary abelian or extraspecial has a nontrivial normal p-subgroup

Statement
Suppose $$P$$ is a fact about::group of prime power order that is neither an fact about::elementary abelian group nor an fact about::extraspecial group. Then, $$\operatorname{Out}(P)$$ has a nontrivial normal $$p$$-subgroup. Equivalently, the $$p$$-fact about::Sylow-core of $$\operatorname{Out}(P)$$, denoted $$O_p(\operatorname{Out}(P))$$, is nontrivial.