The Mathieu groupoid $M_{13}$ has a beautiful construction from a "sliding block puzzle" produced from the projective plane of order $3$; put $12$ labelled counters on $12$ of the $13$ points of the projective plane, and look at all the configurations one can reach by moving a counter on a line with the empty spot onto the empty spot, and swapping the other two counters on that line. This gives a groupoid (like a group, but two moves can only be composed if they are compatible as to where the empty spot is). The marvelous things is that look at all permutations of the $12$ counters the occur with the empty spot in a common place (which now form a group), you get the sporadic group $M_{12}$. For those not familiar with $M_{12}$, note it and $M_{24}$ are the only finite groups that are $5$ transitive (besides the obvious symmetric and alternating groups), though there's a lot more that can be said about their exceptionality!
My questions is as follows: while this construction is excellent, is the structure of $M_{13}$ itself any more interesting than $M_{12}$? If we just see it as a groupoid, the answer seems to be no; any two connected groupoids on the same number of objects with same fundamental group are isomorphic, so such an object existing is not a surprise.
However, this groupoid also comes with a group action by the symmetries of the projective plane, and maybe the groupoid equipped with this action is more of a "surprising" or exceptional fact? It makes me ask the question: for any group $G$ acting on a set $S$, and group $K$, is there always a connected groupoid with objects $S$, automorphism group of any object being $K$, and an action by $G$ that when restricted two the groupoids objects, is the action on $S$? And is it unique? That is, are connected groupoids with a group action classified by more than a the group action and the automorphism group of objects? And if so, is there some clear way in which $M_{13}$ is particularly special, besides involving a sporadic group?
Possibly there is some structure even beyond that of the group action we want to imagine $M_{13}$ having, that demonstrates its exceptionality. I'd appreciate any perspectives on $M_{13}$ that clarify any of these questions, even if they aren't exactly what I have in mind. Even if it turns out only the construction is exceptional (in some sense) and not the resulting object, I'd still be happy enough!