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The group of isomorphism classes of $Z$-bundles on $X$, which is isomorphic to $H^1(X,Z)$, acts on $M(X,G)$ via extension of structure group by the multiplication homomorphism $Z\\times G\\to G$. The group $\\text{Aut}(G)$ also acts on $M(X,G)$ by extension of structure group, and so does the group $\\text{Aut}(X)$ of holomorphic automorphisms via pullback. Finally, $\\mathbb{C}^*$ acts by multiplying the Higgs field. 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