Analogue of critical subgroup theorem for finite solvable groups

Statement

Suppose $G$ is a Finite solvable group (?). Then, there exists a Characteristic subgroup (?) $K$ of $G$ satisfying the following two conditions:

1. $[K,K]\leq \Phi (K)\leq Z(K)$ (the derived subgroup is contained in the Frattini subgroup, which in turn is contained in the center): in other words, $K$ is a Frattini-in-center group (?).
2. $C_{G}(K)=Z(K)$: in other words, $K$ is a 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.