Collection of groups satisfying a universal congruence condition

Definition
Suppose $$\mathcal{S}$$ is a finite collection of finite $$p$$-groups, groups of prime power order for the prime $$p$$. We say that $$\mathcal{S}$$ satisfies a universal congruence condition if the following equivalent conditions are satisfied by $$\mathcal{S}$$:


 * 1) For any finite $$p$$-group $$P$$ that contains a subgroup isomorphic to an element of $$\mathcal{S}$$, the number of subgroups of $$P$$ isomorphic to elements of $$\mathcal{S}$$ is congruent to $$1$$ modulo $$p$$.
 * 2) For any finite $$p$$-group $$P$$ that contains a subgroup isomorphic to an element of $$\mathcal{S}$$, the number of normal subgroups of $$P$$ isomorphic to elements of $$\mathcal{S}$$ is congruent to $$1$$ modulo $$p$$.
 * 3) For any finite $$p$$-group $$Q$$ and any normal subgroup $$P$$ of $$Q$$ such that $$P$$ contains a subgroup isomorphic to an element of $$\mathcal{S}$$, the number of normal subgroups of $$Q$$ isomorphic to elements of $$\mathcal{S}$$ and contained in $$P$$ is congruent to $$1$$ modulo $$p$$.
 * 4) For any finite $$p$$-group $$P$$ that contains a subgroup isomorphic to an element of $$\mathcal{S}$$, the number of p-core-automorphism-invariant subgroups of $$P$$ isomorphic to elements of $$\mathcal{S}$$ is congruent to $$1$$ modulo $$P$$.
 * 5) For any finite group $$G$$ containing a subgroup isomorphic to an element of $$\mathcal{S}$$, the number of subgroups of $$G$$ isomorphic to an element of $$\mathcal{S}$$ is congruent to $$1$$ modulo $$p$$.

Weaker properties

 * Stronger than::Collection of groups satisfying a strong normal replacement condition
 * Stronger than::Collection of groups satisfying a weak normal replacement condition

Threshold values
This lists threshold values of $$k$$: the largest value of $$k$$ for which the collection of $$p$$-groups of order $$p^k$$ satisfying the stated condition satisfies a universal congruence condition. The nature of all these is such that the universal congruence condition is satisfied for all smaller $$k$$ but for no larger $$k$$. We use between $$a$$ and $$b$$ to mean that the value is at least $$a$$ and at most $$b$$.