Congruence condition on number of subgroups of given order and bounded exponent in class two group for odd prime

Statement
Suppose $$p$$ is an odd prime number and $$P$$ is a group of prime power order for the prime $$p$$. Suppose, further, that $$P$$ is a group of nilpotency class two. Suppose $$P$$ has a subgroup $$H$$ of order $$p^k$$ and exponent $$p^d$$.

Then, the number of subgroups of $$P$$ of order exactly $$p^k$$ and exponent at most $$p^d$$ is congruent to 1 mod $$p$$.

In other words, the collection of groups of a given order and a given bound on their exponent satisfies a congruence condition within the world of groups of nilpotency class two.

Related facts

 * Congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group
 * Mann's replacement theorem for subgroups of prime exponent