Center not is intermediately powering-invariant

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 $$H$$.

Opposite facts

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

Proof
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.