Characteristic subgroup of abelian group is quotient-powering-invariant

Statement
Suppose $$G$$ is an abelian group and $$H$$ is a characteristic subgroup of $$G$$ (in other words, $$H$$ is a characteristic subgroup of abelian group). Then, $$H$$ is a quotient-powering-invariant subgroup of $$G$$: if $$G$$ is powered over a prime $$p$$ (i.e., every element of $$G$$ has a unique $$p^{th}$$ root), so is the quotient group $$G/H$$.

Facts used

 * 1) uses::Characteristic subgroup of abelian group is powering-invariant
 * 2) uses::Powering-invariant and central implies quotient-powering-invariant

Proof
The proof follows directly from Facts (1) and (2).