Dear Henry, I did not mean this to be a serious suggestion; but I found this unbelievably funny.

Yes, you are right, my explanation is incorrect. I edited the answer.

The map from the fiber product onto $S$ is smooth, hence open

Dear Chris, yes, I think I can show that one of the classes vanishes if and only if the other vanishes. Also, the two classes have the same orders; this gives a strong indication that they should be coincide up to sign.

To Fernando: you are right; but I am assuming that $\overline G$ is locally connected, or, equivalently, that is a Lie group, so $G$ is discrete. I edited the question to reflect this.

homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf

This is off-topic here, as this website as dedicated to question that are relevant in mathematical research. Please read the FAQ, and try posting at math.stackexchange.com .

A good introduction to reflexive sheaves is "Stable reflexive sheaves", by Hartshorne, Mathematische Annalen (1980).

A covering map $f\colon X \to Y$ is a local diffeomorphism; a form on $X$ around a point $p$ gives a form on $Y$ around $f(p)$. If you have a form on $X$, for each point $q$ of $Y$ sum over the form you get from each point of $f^{-1}(q)$; the forms you get glue together to a global form on $Y$.