A Prym-Narasimhan-Ramanan construction of principal bundle fixed points

Let $X$ be a compact Riemann surface and $G$ be a connected reductive complex Lie group with centre $Z$. Consider the moduli space $M(X,G)$ of polystable principal holomorphic $G$-bundles on $X$. There is an action of the group $H^1(X,Z)$ of isomorphism classes of $Z$-bundles over $X$ on $M(X,G)$ induced by the multiplication $Z\times G\to G\mathrm{.}$ Let $\mathrm{\Gamma }$ be a finite subgroup of $H^1(X,Z)$. Our goal is to find a Prym–Narasimhan–Ramanan-type construction to describe the fixed points of $M(X,G)$ under the action of $\mathrm{\Gamma }$. A main ingredient in this construction is the theory of twisted equivariant bundles on an étale cover of $X$ developed in arXiv:2208.0902(2).