Normal subgroup acting on a set I am trying to solve the following problem:
Let $G$ be a group acting on a set $X$ and let $S \lhd G$. Determine the necessary and sufficient conditions so that there exists an action of $G/S$ on $X$ such that $\overline{a}.x=a.x$ for all $a \in G$ and $x \in X$.
I am pretty much stuck with this exercise, I would appreciate any hints.

Checking conditions of  Group action  does not say much as it is natural..
$\overline{ab}\cdot x=ab\cdot x=a\cdot(b\cdot x)$
$\bar{a}\cdot(\bar{b}\cdot x)=\bar{a}\cdot(b\cdot x)=a\cdot(b\cdot x)$
All that you need to see is does the map makes sense??
I mean is it well defined??
