Subgroup that is divisible by a set of primes in the whole group

Definition
Suppose $$G$$ is a group, $$H$$ is a subgroup of $$G$$, and $$\pi$$ is a set of primes. We say that $$H$$ is $$\pi$$-divisible in $$G$$ if any any $$h \in H$$ and $$p \in \pi$$, there exists $$x \in G$$ such that $$x^p = h$$.

Note importantly that we are only guaranteed the existence of $$p^{th}$$ roots, and not necessarily the existence of $$(p^2)^{th}$$ or higher roots for $$p \in \pi$$. If $$H = G$$, this distinction is not relevant, but it can become relevant when $$H$$ is a proper subgroup of $$G$$.

Facts

 * If $$G$$ is a $$\pi$$-divisible group, then every subgroup of $$G$$ is $$\pi$$-divisible in $$G$$.
 * If $$H$$ is a $$\pi$$-divisible group, then it is $$\pi$$-divisible in $$G$$ regardless of $$G$$.
 * For an ascending chain of subgroups, each of which is $$\pi$$-divisible in its successor, the union is a $$\pi$$-divisible group.