Powering-injectivity is central extension-closed

Statement
Suppose $$G$$ is a group and $$H$$ is a central subgroup of $$G$$. Suppose $$p$$ is a prime number such that:


 * $$H$$ is $$p$$-powering-injective, i.e., $$x \mapsto x^p$$ is injective from $$H$$ to itself.
 * The quotient group $$G/H$$ is $$p$$-powering-injective, i.e., $$x \mapsto x^p$$ is injective from $$G/H$$ to itself.

Then, the whole group $$G$$ is $$p$$-powering-injective.

Related facts

 * Divisibility is central extension-closed
 * Powering is central extension-closed

Proof
Given: A group $$G$$, a prime number $$p$$. A central subgroup $$H$$ of $$G$$ such that in both $$H$$ and $$G/H$$ viewed separately, the map $$x \mapsto x^p$$ is injective. Two elements $$a,b \in G$$ with $$a^p = b^p$$.

To prove: $$a = b$$.

Proof: Let $$\pi:G \to G/H$$ be the quotient map.