Finite-p-potentially fully invariant subgroup

Definition
Suppose $$G$$ is a group of prime power order, i.e., a finite $$p$$-group for some prime number $$p$$. A subgroup $$H$$ of $$G$$ is termed finite-p-potentially fully invariant in $$G$$ if there exists a finite $$p$$-group $$K$$ containing $$G$$ such that $$H$$ is a defining ingredient::fully invariant subgroup of $$K$$.