Powering-invariant subgroup of abelian group

Definition
Suppose $$G$$ is an group and $$H$$ is a subgroup of $$G$$. We say that $$H$$ in $$G$$ is a powering-invariant subgroup of abelian group if the following equivalent conditions are satisfied:


 * 1) $$G$$ is an abelian group and $$H$$ is a powering-invariant subgroup of $$G$$.
 * 2) $$G$$ is an abelian group and $$H$$ is a quotient-powering-invariant subgroup of $$G$$.

Non-examples
Group of integers in group of rational numbers is the standard non-example.