Finite-pi-potentially verbal subgroup

Definition
Suppose $$K$$ is a finite group and $$H$$ is a subgroup of $$K$$. We say that $$H$$ is a finite-pi-potentially verbal subgroup of $$K$$ if the following holds: There exists a group $$G$$ containing $$K$$ such that all prime factors of the order of $$G$$ also divide the order of $$K$$ and $$H$$ is a verbal 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 properties

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