Intermediately strictly characteristic subgroup

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Definition

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 strictly characteristic subgroup (also called distinguished subgroup) of $K$. Here, a strictly characteristic subgroup is a subgroup that is invariant under all surjective endomorphisms.

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Intermediately fully invariant subgroup	fully invariant in every intermediate subgroup
Homomorph-containing subgroup	contains every homomorphic image in the whole group			|FULL LIST, MORE INFO
Prehomomorph-contained subgroup	contained in every subgroup having it as a homomorphic image
Normal subgroup having no nontrivial homomorphism to its quotient group	no nontrivial homomorphism to its quotient group			|FULL LIST, MORE INFO

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Intermediately characteristic subgroup	characteristic in every intermediate subgroup
Strictly characteristic subgroup
Characteristic subgroup