Intermediately normality-large subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed intermediately normality-large if it satisfies the following equivalent conditions:

1. It is normality-large in every intermediate subgroup
2. It does not occur as a retract of any subgroup properly containing it.

Definition with symbols

A subgroup $H$ of a group $G$ is termed intermediately normality-large in $G$ if it satisfies the following equivalent conditions:

1. For any subgroup $K$ of $G$ containing $H$, and any normal subgroup $N$ of $K$ such that $N\cap H$ is trivial, $N$ must be trivial
2. If $K$ is a subgroup of $G$ containing $H$, and $\alpha$ is a retraction from $K$ to $H$ (i.e. a surjective homomorphism that restricts to the identity map on $H$), then $K=H$.