Normal subset-conjugacy-closed subgroup

Symbol-free definition
A subgroup of a group is termed a normal subset-conjugacy-closed subgroup if it is a defining ingredient::normal subset-conjugacy-determined subgroup in itself relative to the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a normal subset-conjugacy-closed subgroup in $$G$$ if for any two defining ingredient::normal subsets $$A,B$$ of $$H$$ such that there exists $$g \in G$$ with $$gAg^{-1} = B$$, then $$A = B$$.

Stronger properties

 * Weaker than::Direct factor
 * Weaker than::Central factor
 * Weaker than::Central subgroup
 * Weaker than::Cocentral subgroup
 * Weaker than::Retract
 * Weaker than::Subset-conjugacy-closed subgroup

Weaker properties

 * Stronger than::WNSCC-subgroup