[Sent here from Math.StackExchange by suggestion of an user.]

By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. Butz. and I. Moerdijk. Representing topoi by topological grupoids. Journal of Pure and Applied Algebra 130, 223-235, 1998) that topoi "with enough points" admit actually a representation as classifying topoi of topological groupoids.

Now my question is the following: take a well-known topos, as the étale topos for a scheme. This is the classifying topos of a localic groupoid, acrtually a topological groupoid since it is "coherent" and thus has enough points by Deligne's theorem; but which groupoid is this in concrete? Do you know if someone has ever investigated that? Thank you in advance.

P.S.: I have been suggested to look at the proof and try to reconstruct the particular case, which I am going to try.