Distinguished set of coset representatives

Definition
Let $$G$$ be a group and $$H$$ be a subgroup of $$G$$. A left transversal $$T$$ for $$H$$ in $$G$$ (i.e., a choice of one element from each left coset of $$H$$ in $$G$$) is termed a distinguished set of coset representatives if $$hth^{-1} \in T$$ for any $$h \in H, t \in T$$.

An analogous definition holds for right coset representatives.

A subgroup of a group need not possess a distinguished set of coset representatives. In fact, a subgroup possesses a distinguished set of coset representatives if and only if it is a subset-conjugacy-closed subgroup.