Notions of groups acting on groups

Question

Let $G$ be a group acting on a set $S$, by means of $(g,s)\mapsto s^g$. If $S$ is itself also a group, then it is natural to impose the further condition that $(st)^g=s^gt^g$. This seems to be the standard notion of a group acting on a group, and the action of a group on itself by conjugation is a central example. But, denoting the group action of $G$ on $S$ by $(g,s)\mapsto g\cdot s$, what about the following natural condition instead: $g\cdot (st)=(g\cdot s) t$ ? The canonical action of a group on itself satisfying this condition. Is there a standard name for such an action? If so, a reference will be appreciated. If not, is there any compelling reason why this notion is not worthy ?