Finite-pi-potentially characteristic subgroup

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

When $$K$$ is a group of prime power order, this is termed finite-p-potentially characteristic subgroup.

Stronger properties

 * Weaker than::Finite-pi-potentially verbal subgroup
 * Weaker than::Finite-pi-potentially fully invariant subgroup
 * Weaker than::Central subgroup of finite group
 * Weaker than::Cyclic normal subgroup of finite group (cyclic normal subgroup of finite group)

Weaker properties

 * Stronger than::Normal subgroup of finite group