  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 $Z\times G\to G.$ Let $\Gamma$ 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 $\Gamma$. A main
ingredient in this construction is the theory of twisted equivariant bundles on
an \'etale cover of $X$ developed in arXiv:2208.0902(2).
