Collection of groups satisfying a property-conditional congruence condition

Definition
Suppose $$p$$ is a prime number and $$\mathcal{S}$$ is a collection of finite $$p$$-groups. Suppose $$\alpha$$ is a property of finite $$p$$-groups. We say that $$\mathcal{S}$$ satisfies a property-conditional congruence condition for property $$\alpha$$ if, for any group $$P$$ satisfying property $$\alpha$$, the number of subgroups of $$P$$ isomorphic to elements of $$\mathcal{S}$$ is either zero or congruent to $$1$$ modulo $$p$$.

When $$\alpha$$ is the property of being any finite $$p$$-group, we say that $$\mathcal{S}$$ is a collection of groups satisfying a universal congruence condition.

Examples
Also see the examples in collection of groups satisfying a universal congruence condition.