Finite-pi-potentially fully invariant subgroup

Definition
Let $$K$$ be a group, and $$H$$ be a subgroup of $$K$$. We say that $$H$$ is a finite-pi-potentially fully invariant subgroup of $$K$$ if there exists a finite group $$G$$ containing $$K$$ such that every prime factor of the order of $$G$$ also divides the order of $$K$$, and such that $$H$$ is a fully invariant subgroup of $$G$$.

Stronger properties

 * Weaker than::Central subgroup of finite group
 * Weaker than::Cyclic normal subgroup of finite group
 * Weaker than::Homocyclic normal subgroup of finite group
 * Weaker than::Finite-pi-potentially verbal subgroup

Weaker properties

 * Stronger than::Finite-pi-potentially characteristic subgroup