Characteristic subgroup of abelian group implies powering-invariant

Statement
Suppose $$G$$ is an abelian group and $$H$$ is a characteristic subgroup of $$G$$. Then, $$H$$ is a powering-invariant subgroup of $$G$$: for any prime number $$p$$ such that every element of $$G$$ has a unique $$p^{th}$$ root, every element of $$H$$ also has a unique $$p^{th}$$ root in $$H$$.

Related facts

 * Characteristic not implies powering-invariant
 * Characteristic subgroup of abelian group implies intermediately powering-invariant

Facts used

 * 1) uses::Abelian implies universal power map is endomorphism

Proof idea
The idea is to use Fact (1), and the powering, to show that the $$p^{th}$$ power map is an automorphism, hence so is its inverse (the $$p^{th}$$ root map), and hence, because the subgroup is characteristic, it is invariant under the map.