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
H1(X,Z) of isomorphism classes of Z-bundles over X on M(X,G) induced
by the multiplication Z×G→G. Let Γ be a finite subgroup of
H1(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 Γ. 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).Comment: 52 pages. In this version we have substantially restructured the
content of the pape