Subgroup having a left transversal that is also a right transversal

Statement
Suppose $$H$$ is a subgroup of a group $$G$$. We say that $$H$$ is a subgroup having a left transversal that is also a right transversal if there exists a subset $$S$$ of $$G$$ such that $$S$$ is a left transversal of $$H$$ and is also a right transversal of $$H$$.

Non-examples
See subgroup need not have a left transversal that is also a right transversal for an example of a subgroup that does not satisfy this property. In the example, the whole group is isomorphic to free group:F2 and the subgroup is free on a countable generating set.