Subgroup having a symmetric transversal

Definition
A subgroup $$H$$ of a group $$G$$ is termed a subgroup having a symmetric transversal if there exists a symmetric subset $$S$$ of $$G$$ such that $$S$$ is a left transversal of $$G$$. Note that for a symmetric subset, being a left transversal is equivalent to being a right transversal, because left and right coset spaces are naturally isomorphic via the bijection induced by the inverse map.