Intermediately strictly characteristic subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed an intermediately strictly characteristic subgroup if for every subgroup $$K$$ of $$G$$ containing $$H$$, $$H$$ is a defining ingredient::strictly characteristic subgroup (also called distinguished subgroup) of $$K$$. Here, a strictly characteristic subgroup is a subgroup that is invariant under all surjective endomorphisms.