Possibilities for a group $G$ that acts faithfully on a set of objects with two orbits? 
A group $G$ acts faithfully on a set $X$ of 5 objects. The action has
  two orbits: one of size 2, and one of size 3. What are the
  possibilities for the group $G$?

I think I should apply the orbit-stabilizer theorem. I believe a main hint is the fact that this group has precisely two orbits but I'm not sure how to use that information. What's a good way to go about listing the possibilities for this group $G$?
 A: First determine the order of the group. Remember that it is a subgroup of $S_{5}$. The size of an orbit must divide the size of the group, which in turn must divide the order of $S_{5}$; so this means that $|G|$ is a multiple of $6$ which also divides $120$. 
On the surface that looks like there are a lot of options, but consider that any $\sigma \in G$ can be decomposed into its' action on each orbit, so really you can consider $G$ a subgroup of $S_{2} \times S_{3}$, which has order $2\cdot 6 = 12$. Notice this also means you can describe the possibilities for $G$ by classifying the transitive subgroups of $S_{2}$ and $S_{3}$.
[EDIT] My hint really only makes two of the three possibilities apparent, namely $\mathbb{Z}_{2} \times \mathbb{Z}_{3}$ (order 6) and $\mathbb{Z}_{2} \times S_{3}$ (order 12). But as noted by Derek Holt, there is actually a third possibility, namely a second group of order 6. I would suggest to consider how $G = S_{3}$ can act on the orbits $\{1,2\}$ and $\{3,4,5\}$ in an appropriate way, by considering the action of the two generators of the group (i.e. consider $S_{3} = \langle \alpha, \beta \mid \alpha^{3} = \beta^{2} = e,\ \beta \alpha \beta = \alpha^{-1} \rangle$).
