Nilpotent-quotient implies subgroup-to-quotient powering-invariance implication

Statement
Suppose $$G$$ is a group and $$H$$ is a normal subgroup of $$G$$ that is a nilpotent-quotient subgroup, i.e., the quotient group $$G/H$$ is a nilpotent group (note that $$G$$ and $$H$$ themselves may or may not be nilpotent). Then, $$H$$ is a normal subgroup satisfying the subgroup-to-quotient powering-invariance implication in $$G$$. Explicitly, this means that if $$p$$ is a prime number such that both $$G$$ and $$H$$ are $$p$$-powered, then so is $$G/H$$.

Facts used

 * 1) uses::Divisibility is inherited by quotient groups
 * 2) uses::Equivalence of definitions of nilpotent group that is torsion-free for a set of primes: We use the equivalence of (1) and (2) within the multi-part equivalence. This says that a nilpotent group has no non-identity elements of order $$p$$ if and only if its $$p^{th}$$ power map is injective.

Proof
Given: Group $$G$$, normal subgroup $$H$$ such that both $$G$$ and $$H$$ are $$p$$-powered for some prime number $$p$$.

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

Proof: