Congruence condition on number of normal subgroups with quotient in a specific variety in a group of prime power order

Statement
Suppose $$\mathcal{V}$$ is a subvariety of the variety of groups, $$p$$ is a prime number, and $$G$$ is a finite p-group (not necessarily in $$\mathcal{V}$$), i.e., a group of order a power of $$p$$. Suppose $$p^r$$ is a power of $$p$$ that divides the order of $$G$$.

Let $$\mathcal{S}$$ be the collection of normal subgroups $$H$$ of $$G$$ of order $$p^r$$ such that $$G/H \in \mathcal{V}$$. Then, either $$\mathcal{S}$$ is empty or the size of $$\mathcal{S}$$ is congruent to 1 modulo $$p$$.

Similar facts

 * Congruence condition on number of subgroups of given prime power order
 * Congruence condition on number of ideals with quotient in a specific variety in a nilpotent ring
 * Congruence condition on number of ideals of given prime power order in nilpotent ring

Facts used

 * 1) uses::Congruence condition on number of subgroups of given prime power order

Proof
, mostly direct from Fact (1) and the observation that for the quotient to be in a specific variety is equivalent to the subgroup containing a specific characteristic subgroup dependent on the variety, so we can pass to the quotient and count subgroups of the relevant order.