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

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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 Characteristic subgroup (?)s (see also 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.

Related facts

Stronger facts

Opposite facts

Similar facts

Proof

Case p = 2

Further information: central product of D8 and Z8

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: