In 1975 Narasimhan and Ramanan studied the action of a line bundle $L$ of finite order on the moduli space of vector bundles of fixed rank and degree via tensorization. They proved that fixed points are pushforwards of vector bundles of lower degree over an Étale cover of $X$ determined by $L$. We extend this study to $G$-Higgs bundles for $G$ reductive and consider the more general action of a finite group $\Gamma$. The action of an element in $\Gamma$ involves "tensorization by a line bundle", extension of structure group by an automorphism of $G$ and rescaling the Higgs field. We describe the fixed points in terms of twisted equivariant bundles over some Galois covers of $X$, which is the counterpart of the result by Narasimhan and Ramanan. Possible applications include producing examples of branes to test mirror symmetry.