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.

Hint.  Think about what it would mean for the given action ($\bar{a}\cdot x = a\cdot x$) to be well-defined.  That means that, whenever $\bar{a}=\bar{b}$, for $a,b\in G$, you must have $\bar{a}\cdot x = \bar{b}\cdot x$, for all $x$ in $X$.
