Congruence condition fails for number of characteristic subgroups in group of prime power order

Statement
Let $$p$$ be any prime number.

It is possible to have the following situation: $$P$$ is a finite p-group of order $$p^k$$, and there exists $$r$$ with $$0 \le r \le k$$ such that the number of fact about::characteristic subgroups (see also fact about::characteristic subgroup of group of prime power order) of $$P$$ of order $$p^r$$ is a nonzero number that is not congruent to 1 mod $$p$$.

Stronger facts

 * Congruence condition fails for number of characteristic subgroups in abelian group of prime power order

Opposite facts

 * Congruence condition on number of subgroups of given prime power order tells us that the opposite is true if we are looking at all subgroups or at all normal subgroups.

Similar facts

 * Congruence condition fails for number of normal subgroups of given prime power order: Note that this states that the number of normal subgroups of a given prime power order may be a nonzero number not congruent to 1 modulo the prime, but to construct a counterexample, we need to move to an ambient finite group that is not itself a p-group.

Case $$p = 2$$
Consider $$G$$ to be the central product of D8 and Z8, a group of order 32 (GAP ID: (32,38)). $$G$$ has two characteristic subgroups of order 8:


 * The center, which is isomorphic to cyclic group:Z8.
 * An isomorph-free subgroup isomorphic to the quaternion group.