Complemented normal implies quotient-powering-invariant

Statement
Suppose $$G$$ is a group and $$H$$ is a complemented normal subgroup of $$G$$ (i.e., there exists a permutable complement to $$H$$ in $$G$$, i.e., $$G$$ is an internal semidirect product involving $$H$$). Then, $$H$$ is a quotient-powering-invariant subgroup of $$G$$: for any prime number $$p$$ such that $$G$$ is powered over $$p$$ (i.e., every element of $$G$$ has a unique $$p^{th}$$ root), $$G/H$$ is also powered over $$p$$.

Facts used

 * 1) uses::Complemented normal implies endomorphism kernel
 * 2) uses::Endomorphism kernel implies quotient-powering-invariant

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

Hands-on proof
Given: A group $$G$$, a normal subgroup $$H$$ of $$G$$ with permutable complement $$K$$. $$G$$ is powered over a prime number $$p$$.

To prove: $$G/H$$ is powered over $$p$$.

Proof: