Injection of a finite hyperbolic isometry group into a mapping class group

If a finite group $G$ acts faithfully by orientation-preserving isometries on a closed hyperbolic surface $S$ of genus at least two, then $G\hookrightarrow \mathrm{Mod}(S)$. An isometry isotopic to the identity induces an inner automorphism of the surface group; its suitable lift commutes with every deck transformation and fixes their limit set, forcing the lift and the original isometry to be the identity.