Endomorphism image implies divisibility-closed

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., endomorphism image) must also satisfy the second subgroup property (i.e., divisibility-closed subgroup)
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Statement

Suppose $G$ is a group and $H$ is a subgroup of $G$ that is an endomorphism image of $G$, i.e., there exists an endomorphism $\sigma$ of $G$ such that $\sigma (G)=H$. Then, $H$ is a divisibility-closed subgroup of $G$, i.e., if $n$ is a natural number such that every element of $G$ has a $n^{th}$ root (not necessarily unique) in $G$, then every element of $H$ has a $n^{th}$ root (not necessarily unique) in $H$.

Facts used

1. Divisibility is inherited by quotient groups

Proof

Proof idea

The proof idea is to take an inverse image under the endomorphism, take the $n^{th}$ root, and then take the image again. More abstractly, we can simply use Fact (1).