Congruence condition on factorization of Hall numbers

Statement
Suppose $$G$$ is a finite solvable group and $$\pi$$ is a set of primes. Then, the number of $$\pi$$-Hall subgroups of $$G$$ can be expressed as a product of factors, each of which is congruent to 1 modulo $$p$$ for some $$p \in \pi$$ (where the $$p$$ could differ across factors).

Related facts

 * Congruence condition on Sylow numbers
 * Congruence condition on number of subgroups of given prime power order