I am looking at $S_{4}$, the set of permutations of the integers $\{1,2,3,4\}$, which has $4!$ elements. I am trying to describe all ways in which $S_{4}$ acts on a set with two elements, say $S = \{a,b\}$. I know that the only subgroup of order $12$ of the symmetric group $S_{4}$ is the alternating group $A_{4}$, but I do not know how to specifically apply this to this problem.
Approach: Let f: $S_{4}$ -> $S_{2}$. |im f| = 2 or 1, so then |ker f| = |$S_{4}$|/|im f| = 24/2 = 12 or = 24. im f is a subgroup of $S_{4}$ so im f is $A_{4}$ or $S_{4}$. I am not sure what to do next.