Assume that $\Gamma$ is a group with neutral element $e$. We associate to $\Gamma$ the following groupoid $G$:


$G=\Gamma \times \Gamma,\;\;\;G^{(0)}=\Gamma \times \{0\},\;\;s(a,b)=(a,e),\;\;\; r(a,b)=(ba, e)$


If $\phi:\Gamma\_{1}\to \Gamma\_{2}$ is a group isomorphism, then $\tilde{\phi}:G\_{1} \to G\_{2}$ with $\tilde{\phi}(a,b)=(\phi(a), \phi(b))$ is a groupoid isomorphism. So isomorphism groups give us isomorphic groupoids.
Now we ask the converse:



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> Are there two non isomorphic groups $\Gamma\_{1}, \Gamma\_{2}$ such that the corresponding groupoids $G\_{1}, G\_{2}$ are isomorphic.
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Note that a groupoid isomorphism between $G\_{1}, G\_{2}$ does not necessarily come from a group isomorphism between $\Gamma\_{1}, \Gamma\_{2}$, as constructed above. An easy example can be provided by $\Gamma\_{1}=\Gamma\_{2}=\mathbb{Z}/2\mathbb{Z}$. 


This situation is a motivation for the above question.



Groupoid isomorphism vs. group isomorphism
Look at the group $Aut(x)$ where $x$ is an object of the groupoid. You find that this group is trivial. Therefore your groupoid is THE groupoid with trivial automorphism groups and its isomorphy class only depends on the cardinality of the set of objects.
Hence any two groups of the same cardinality will give isomorphic groupoids


