Intermediately local powering-invariant subgroup

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

Facts

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