Subset-conjugacy-closed subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed subset-conjugacy-closed in $$G$$ if it satisfies the following equivalent conditions:


 * 1) For any subsets $$A,B$$ of $$H$$, such that there exists $$g \in G$$ with $$gAg^{-1} = B$$, there exists $$h \in H$$ such that $$hah^{-1} = gag^{-1}$$ for all $$a \in A$$.
 * 2) $$H$$ is a subset-conjugacy-determined subgroup of itself with respect to $$G$$, i.e., the fusion for subsets of $$H$$ in $$G$$, is contained in $$H$$.
 * 3) $$H$$ possesses a defining ingredient::distinguished set of coset representatives in $$G$$: In other words, there is a set $$T$$ of left coset representatives of $$H$$ in $$G$$ such that $$hth^{-1} \in T$$ for all $$h \in H, t \in T$$.