Finite-potentially fully invariant subgroup

Definition
A subgroup $$H$$ of a finite group $$G$$ is termed a finite-potentially fully invariant subgroup if there exists a finite group $$K$$ containing $$G$$ such that $$H$$ is a fully invariant subgroup of $$K$$.

Stronger properties

 * Weaker than::Fully invariant subgroup of finite group
 * Weaker than::Verbal subgroup of finite group
 * Weaker than::Finite-potentially verbal subgroup
 * Weaker than::Finite-pi-potentially verbal subgroup
 * Weaker than::Finite-pi-potentially fully invariant subgroup