Normal subgroup of nilpotent group satisfies the subgroup-to-quotient powering-invariance implication

Statement
Suppose $$G$$ is a nilpotent group and $$H$$ is a normal subgroup of $$G$$. Then, $$H$$ is a normal subgroup satisfying the subgroup-to-quotient powering-invariance implication. Explicitly, if $$p$$ is a prime number such that both $$G$$ and $$H$$ are $$p$$-powered, so is the quotient group $$G/H$$.

Facts used

 * 1) uses::Normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication
 * 2) uses::Nilpotent-quotient implies subgroup-to-quotient powering-invariance implication

Proof using Fact (1)
For the proof using Fact (1), note that a nilpotent group equals its own hypercenter, so any normal subgroup is contained in the hypercenter. Thus, Fact (1) applies, and we get the result.

Proof using Fact (2)
For the proof using Fact (2), note that nilpotency is quotient-closed, so the quotient group is nilpotent, hence Fact (2) applies, and we get the result.