2-regular group action

Definition
Let $$G$$ be a group and $$S$$ be a set with a group action of $$G$$ on $$S$$. The action is 2-regular if the following is true: for any $$a \ne b \in S$$ and any $$c \ne d \in S$$, there is a unique $$g \in G$$ such that $$g \cdot a = c, g \cdot b = d$$.

Facts

 * 2-regular group action implies elementary Abelian regular normal subgroup