Finite p-group that is not characteristic in any finite p-group properly containing it

History

The existence of such groups, along with some examples and important facts, was established in a paper by Bettina Wilkens.

Definition

Suppose $p$ is a prime number and $P$ is a finite $p$-group, i.e., a group of prime power order where the underlying prime is $p$. We say that $P$ is a finite p-group that is not characteristic in any finite p-group properly containing it if, for any finite $p$-group $Q$ containing $P$, $P$ is not a characteristic subgroup (i.e., characteristic subgroup of group of prime power order) of $Q$.

If $P \le Q$ and $P$ satisfies this property, and $P$ is proper in $Q$, then $P$ is not a p-finite-potentially characteristic subgroup of $Q$.