A set $X$ is a $G$-set of a group $G$ if there exists a group action $\phi:G\to Sym(X)$.
How many such sets exist (up to isomorphism)?
I know that $S_3$ has three conjugacy classes (one with $id$, one with transpositions, and one with cycles of length $3$) and thus I concluded that there must be three transitive G-sets, namely coset spaces of those classes, with $6,3,2$ elements. Is this the correct answer? Should I actually consider all subgroups of $S_3$? In that case, there are $6$ subgroups and then I would have more G-sets.
A transitive $G$-set $X$ is defined up to equivalence by the conjugacy class of a point stabiliser $H\le G$ (prove by showing that the action of $G$ on $X$ is equivalent to the natural action of $G$ on the cosets of $H$).
$S_3$ has four conjugacy classes of subgroups, with representatives of orders $1,2,3,6$ so up to equivalence $S_3$ has four $G$-sets