Endomorphism image implies powering-invariant

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 powering-invariant subgroup of $$G$$, i.e., if $$G$$ is powered over a prime $$p$$ (i.e., every element of $$G$$ has a unique $$p^{th}$$ root, so is $$H$$.

Facts used

 * 1) uses::Endomorphism image implies divisibility-closed
 * 2) uses::Divisibility-closed implies powering-invariant

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