Endomorphism image implies divisibility-closed

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) uses::Divisibility is inherited by quotient groups

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).