Congruence condition on number of subgroups of given prime power order satisfying any given property weaker than normality

Statement
Let $$G$$ be a finite group and $$p$$ be a prime number. Let $$p^r$$ be a power of $$p$$ dividing the order of $$G$$. Further, let $$\alpha$$ be any subgroup property weaker than normality: in other words, any normal subgroup of a group always satisfies $$\alpha$$ in the group. Then, the number of subgroups of $$G$$ satisfying property $$\alpha$$ is congruent to $$1$$ modulo $$p$$.

Related facts

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