It is written in Wikipedia <http://en.wikipedia.org/wiki/Groupoid>, that any *connected groupoid* $A\rightrightarrows X$ is isomorphic to an *action groupoid* $G\ltimes X$ coming from a transitive action of some group $G$ on $X$. I do not understand how to construct such a group $G$, and would be grateful for an explanation or a reference.


I think that in general one cannot recover $G$ from the action groupoid $G\ltimes X$. Indeed, if $G$ acts *simply transitively* on $X$, then the action groupoid is given by the equivalence relation $X\times X$ on $X$, hence does not depend on $G$, provided that ${\rm Card}(G)={\rm Card}(X)$. Is this correct?


This question is a version of [my question at Math Stack Exchange](https://math.stackexchange.com/questions/361254/connected-groupoids-and-action-groupoids)
to which I got no answers.



Connected groupoids and action groupoids
Here's what I wrote on Math Stack Exchange:


A connected groupoid *A* can be written as an action groupoid for many different groups *G*. All the groupoid determines is *H*, the group of automorphisms of any object in the groupoid, and the index of *H* in *G*, which is the cardinality of the set of objects of the groupoid. And any group *G* with subgroup *H* of the correct index, the action of *G* on the set of cosets of *H* has action groupoid isomorphic to *A*.


This is to be expected, because if we think of *H* as a one object groupoid and of *X* as an indiscrete groupoid (the set of objects is *X*, and there is a unique morphism between any pair of objects) then the original groupoid *A* is isomorphic to the product *H* × *X*, so the isomorphism class of *A* depends only on the group *H* and the cardinality of *X*.


As an extreme example of this, let *G* act on itself by translation and take the action groupoid. This has set of objects *G* and a unique morphism between every pair of elements. Notice all traces of the group structure of *G* are gone: the isomorphism class of this indiscrete groupoid only depends on the cardinality of *G*.


UPDATE 2: Here is a sloganized answer to the question: the **equivalence** class of a connected groupoid *A* is determined by the isomorphism class of the group *H* = Aut(*x*0); the **isomorphism** class of a category is given by the data of its equivalence class plus the number of isomorphic copies of each object in a skeleton.


UPDATE: Here is a proof of the claims above "UPDATE 2".


**Claim 1**: *A* is isomorphic to *H* × *X*.


*Proof.* Choose an object *x*0 of *A*, identify *H* with Aut(*x**0*) and choose arbitrary morphisms *a**x* : *x**0* → *x*. The isomorphism *H* × *X* → *A* is the identity on objects and sends a morphism (*h*, *u*) : *x* → *y* to the morphism *a**y* *h a**x*-1. (Here *u* is the unique morphism in *X* from *x* to *y*.) The inverse *A* → *H* × *X* sends a morphism *a* : *x* → *y* to (*a**y*-1 *a a**x*, *u*) --same *u* as above.


**Claim 2**: For any group *G* with a subgroup *H* of index |*X*|, the action groupoid of *G* acting on the set *G/H* of cosets is isomorphic to *A*. 


*Proof.* Both groupoids are isomorphic to *H* × *X*.


