Powering is central extension-closed

Statement
Suppose $$G$$ is a group, $$p$$ is a prime number, and $$H$$ is a central subgroup of $$G$$ such that the following are true:


 * 1) $$H$$ is powered over $$p$$.
 * 2) $$G/H$$ is powered over $$p$$.

Then, $$G$$ is also powered over $$p$$.

Related facts

 * Central implies normal satisfying the subgroup-to-quotient powering-invariance implication

Facts used

 * 1) uses::Divisibility is central extension-closed
 * 2) uses::Powering-injectivity is central extension-closed

Proof
Fact (1) gives the existence of $$p^{th}$$ roots in $$G$$, whereas Fact (2) gives their uniqueness.