Center not is intermediately powering-invariant

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) does not always satisfy a particular subgroup property (i.e., intermediately powering-invariant subgroup)
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Statement

It is possible to have a group $G$ such that the center $Z(G)$ is not an intermediately powering-invariant subgroup of $G$, i.e., there exists an intermediate subgroup $H$ of $G$ such that $Z(G)$ is not a powering-invariant subgroup of $H$.

Related facts

Opposite facts

• Upper central series members are intermediately local powering-invariant in nilpotent group

Proof

Further information: Amalgamated free product of two copies of group of rational numbers over group of integers

Let $G$ be the group $\mathbb {Q} *_{\mathbb {Z} }\mathbb {Q}$ and $H$ be the first factor $\mathbb {Q}$ as a subgroup of $G$. Then, $Z(G)$ inside $H$ is like Z in Q, and it is not powering-invariant.