Congruence condition on number of cyclic subgroups of small prime power order

Statement
Suppose $$p$$ is any prime number and $$0 \le k \le 5$$. Suppose $$P$$ is a group of prime power order where the prime is $$p$$. Then, the number of cyclic subgroups of $$P$$ of order $$p^k$$ is congruent to either $$0$$ or $$1$$ modulo $$p$$.

Note that the statement is trivially true for $$p = 2$$, so it suffices to prove it for odd $$p$$.

Facts used

 * 1) uses::Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
 * 2) uses::Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime