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

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup of nilpotent group) must also satisfy the second subgroup property (i.e., normal subgroup satisfying the subgroup-to-quotient powering-invariance implication)
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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. Normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication
2. Nilpotent-quotient implies subgroup-to-quotient powering-invariance implication

Proof

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.