Everything that's been written so far about the classification of finite groupoids reducing to the classification of finite groups is true but, I think, misleading. In order to actually produce a list of finite groups from a finite groupoid $X$ you need to choose a basepoint in each connected component of $X$. For example, to produce $M_{12}$ from $M_{13}$ you need to choose one of $13$ points.

There are various reasons it's undesirable to make such choices, with maybe the most practical one being that they are often unavailable once you introduce extra structure. For example, given a group $G$ you might want to study groupoids equipped with a $G$-action. These are strictly more interesting than disjoint unions of groups equipped with a $G$-action, and the reason is precisely that a $G$-groupoid need not have any $G$-invariant basepoints.

This is in fact an issue in the present example:

The Mathieu groupoid $M_{13}$ is equipped with an action of $\text{SL}_3(\mathbb{F}_3)$ which does not fix any points. When you pick a basepoint and get $M_{12}$ back you lose this action.

Another example of this phenomenon is the following: the configuration space $\text{Conf}_k(\mathbb{R}^2)$ of $k$ ordered points in the plane is naturally an Eilenberg-MacLane space $K(P_k, 1)$, where $P_k$ is the pure braid group. On the other hand, $\text{Conf}_k(\mathbb{R}^2)$ clearly has an action of $S_k$ on it given by permuting points. This action does not fix any basepoints, and so it does not induce an action on $P_k$.

It *does* induce an action on a groupoid equivalent to $P_k$ with $k!$ points given by taking the fundamental groupoid of $\text{Conf}_k(\mathbb{R}^2)$ at a set of basepoints which are setwise fixed under the action of $S_k$. A more complicated choice of basepoints setwise fixed under the action of $S_k$ can be used to give a model of the little disks operad as an operad in groupoids. This operad *cannot* be simplified to an operad in groups, despite the fact that all of the groupoids involved are connected, precisely because of basepoint issues (the basepoints must be chosen not only compatibly with the $S_k$-actions but with operadic composition).

**Edit:** One way to measure the interestingness of the $S_k$ action on $\text{Conf}_k(\mathbb{R}^2)$ is that it does not even fix a basepoint in the homotopy coherent sense: equivalently, there is a corresponding short exact sequence

$$1 \to P_k \to B_k \to S_k \to 1$$

that does not split. On the other hand, as pointed out in the comments, any extension of $\text{SL}_3(\mathbb{F}_3)$ by $M_{12}$ must be trivial.

pointedconnected groupoids. See the appendix in Baez and Shulman's article: arxiv.org/abs/math/0608420, p. 50 and following. – Todd Trimble♦ Feb 27 '13 at 18:19