Let $X$ be an algebraic variety over an algebraically closed field $k$ and let a finite group $G$ act on it so that it acts freely on the generic fibre of the projection $X \to X/G$, so $[k(X):k(X/G)]$ is Galois. Consider group extension $1 \to Gal(k(X)) \to Gal(k(X/G)) \to G \to 1$. I want to understand what the differential on the second page of the corresponding LyndonHochschildSerre spectral sequence (the transgression map) does: $$ tr: H^1(k(X), M)^G \to H^2(G, M) $$ (say, $M=\mu_n$ to speak about something concrete). Here, a class in $H^1(k(X), M)^G$ is a class corresponding to some $\mu_n$torsor, and the transgression map puts into correspondence to this class a class of some (central) group extension of $G$ by $\mu_n$. Can this group extension be described in some direct way, without going through the computation of the transgression (which is described, for example, in a reply to this question)?
