Subgroup having a twisted subgroup as transversal

Definition
Suppose $$H$$ is a subgroup of a group $$G$$. We say that $$H$$ is a subgroup having a twisted subgroup as transversal if there exists a twisted subgroup $$S$$ of $$G$$ that is also a left transversal for $$H$$ in $$G$$, i.e., it intersects every left coset of $$H$$ in $$G$$ at exactly one point. Note that $$S$$ will automatically also be a right transversal for $$H$$ in $$G$$.