Finite-p-potentially characteristic subgroup

Statement

Suppose $p$ is a prime number and $G$ is a finite $p$-group. 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