Powering is central extension-closed

This article gives the statement, and possibly proof, of a group property (i.e., powered group for a set of primes) satisfying a group metaproperty (i.e., central extension-closed group property)
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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. Divisibility is central extension-closed
2. Powering-injectivity is central extension-closed

Proof

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