Number of nth roots of a subgroup is divisible by order of subgroup

Statement
Suppose $$G$$ is a finite group, $$H$$ is a subgroup of $$G$$, and $$n$$ is a positive integer. Consider the set:

$$S := \{ g \in G \mid g^n \in H \}$$

Then, the number of elements of $$S$$ is divisible by the order of $$H$$.

Similar facts

 * Number of nth roots is a multiple of n
 * Number of nth roots of any conjugacy class is a multiple of n

Conjectures

 * Frobenius conjecture on nth roots