Normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication

Statement

Suppose $G$ is a group and $H$ is a normal subgroup contained in the hypercenter of $G$ that is also a powering-invariant subgroup of $G$. Then, $H$ is a quotient-powering-invariant subgroup of $G$.

Facts used

1. Powering-invariant and central implies quotient-powering-invariant
2. Quotient-powering-invariance is quotient-transitive