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

History
This result was proved in a paper by Bettina Wilkens in 2008.

Statement
Suppose $$p$$ is a prime number and $$P$$ is a finite $$p$$-group, i.e., a group of prime power order. Then, there exists a finite $$p$$-group $$Q$$ containing $$P$$ such that for any finite $$p$$-group $$R$$ properly containing $$Q$$, $$Q$$ is not a characteristic subgroup of $$R$$. In other words, $$Q$$ is a fact about::finite p-group that is not characteristic in any finite p-group properly containing it.

Related facts

 * Normal subgroups need not be finite-p-potentially characteristic subgroups.
 * Normal not implies finite-pi-potentially characteristic