Coprime automorphism-faithful characteristic subgroup

Definition
A subgroup $$H$$ of a finite group $$G$$ is termed copime automorphism-faithful characteristic if every automorphism of $$G$$ restricts to an automorphism of $$H$$ (i.e., $$H$$ is a defining ingredient::characteristic subgroup) and if $$K$$ is the kernel of the map:

$$\operatorname{Aut}(G) \to \operatorname{Aut}(H)$$

defined by restriction, then every prime divisor of the order of $$K$$, divides the order of $$G$$.

Stronger properties

 * Weaker than::Critical subgroup

Weaker properties

 * Stronger than::Characteristic subgroup
 * Stronger than::Coprime automorphism-faithful subgroup