Characteristic subgroup of abelian group is quotient-powering-invariant

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup of abelian group) must also satisfy the second subgroup property (i.e., quotient-powering-invariant subgroup)
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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. Characteristic subgroup of abelian group is powering-invariant
2. Powering-invariant and central implies quotient-powering-invariant

Proof

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