Odd-order p-group has coprime automorphism-faithful characteristic class two subgroup of prime exponent

Statement
Suppose $$p$$ is an odd prime and $$P$$ is a finite $$p$$-group (i.e., a group of prime power order). Then, there exists a subgroup $$K$$ of $$P$$ satisfying the following conditions:


 * $$K$$ is a fact about::coprime automorphism-faithful characteristic subgroup of $$P$$: $$K$$ is characteristic in $$P$$ and also fact about::coprime automorphism-faithful subgroup}coprime automorphism-faithful in $$P$$.
 * $$K$$ is a group of nilpotence class two.
 * $$K$$ is a group of prime exponent: the exponent of $$K$$ is $$p$$.

Facts used

 * 1) uses::Thompson's critical subgroup theorem
 * 2) uses::Omega-1 of odd-order p-group is coprime automorphism-faithful
 * 3) uses::Omega-1 of odd-order class two p-group has prime exponent
 * 4) uses::Coprime automorphism-faithful characteristicity is transitive
 * 5) uses::Nilpotence class two is subgroup-closed