Analogue of critical subgroup theorem for finite solvable groups

Statement
Suppose $$G$$ is a fact about::finite solvable group. Then, there exists a fact about::characteristic subgroup $$K$$ of $$G$$ satisfying the following two conditions:


 * 1) $$[K,K] \le \Phi(K) \le Z(K)$$ (the derived subgroup is contained in the Frattini subgroup, which in turn is contained in the center): in other words, $$K$$ is a fact about::Frattini-in-center group.
 * 2) $$C_G(K) = Z(K)$$: in other words, $$K$$ is a fact about::self-centralizing subgroup of $$G$$.
 * 3) $$K$$ is coprime automorphism-faithful in $$G$$: any automorphism of $$G$$ of order relatively prime to the order of $$G$$, that restricts to the identity automorphism on $$K$$, must be the identity automorphism on $$G$$.

This statement is a weaker analogue of Thompson's critical subgroup theorem, that holds for a group of prime power order.