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 $$Q$$.

Facts

 * Sylow subgroup of holomorph of cyclic group of prime-cube order is a finite p-group that is not characteristic in any finite p-group property containing it for odd prime
 * Every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it