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:

Are there two non isomorphic groups $\Gamma_{1}, \Gamma_{2}$ such that the corresponding groupoids $G_{1}, G_{2}$ are isomorphic.

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.