Intermediately powering-invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed an intermediately powering-invariant subgroup if, for any subgroup $$K$$ of $$G$$ containing $$H$$ (i.e., $$H \le K \le G$$), $$H$$ is a defining ingredient::powering-invariant subgroup of $$K$$.

Facts

 * Local powering-invariant subgroup containing the center is intermediately powering-invariant in nilpotent group