Finite-p-potentially characteristic subgroup

Statement
Suppose $$p$$ is a prime number and $$G$$ is a finite $$p$$-group (i.e., a group of prime power order). In other words, $$G$$ is a group of prime power order. A subgroup $$H$$ of $$G$$ is termed a finite-$$p$$-potentially characteristic subgroup if there exists a finite $$p$$-group $$K$$ containing $$G$$ such that $$H$$ is a characteristic subgroup of $$K$$.

Facts

 * Every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it

Relation with other properties
The generalization of this property to finite groups, rather than just finite $$p$$-groups, is the property of being a finite-pi-potentially characteristic subgroup.

Stronger properties

 * Weaker than::Characteristic subgroup of group of prime power order
 * Central subgroup of group of prime power order
 * Cyclic normal subgroup of group of prime power order
 * Homocyclic normal subgroup of group of prime power order

Weaker properties

 * Stronger than::Normal subgroup of group of prime power order