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.