Kernel of a characteristic action on an abelian group

Definition
A subgroup $$H$$ of a group $$G$$ is termed a kernel of a characteristic action on an abelian group if there exists an abelian group $$V$$ and a homomorphism $$\alpha:G \to \operatorname{Aut}(V)$$ with kernel $$H$$, such that $$V$$ is a characteristic subgroup of the semidirect product $$V \rtimes G$$.

Stronger properties

 * Weaker than::Normal subgroup of finite group
 * Normal subgroup of a group having no nontrivial abelian normal $$p$$-subgroup for some prime $$p$$.

Weaker properties

 * Stronger than::Strongly image-potentially characteristic subgroup:
 * Stronger than::Semi-strongly image-potentially characteristic subgroup
 * Stronger than::Image-potentially characteristic subgroup
 * Stronger than::Normal subgroup