Curvature markers distinguishing finite-cover quotients

For a finite isometric group action on a closed Riemannian surface, choose a disk in the free locus whose translates are disjoint. Modify the Riemannian metric there, and identically on its translates, by small supported changes with three distinguished high Gaussian curvature maxima. Give their heights distinct values occurring nowhere else, arrange them as a tight noncollinear cluster, and keep separate clusters far apart. Such markers can be made by local conformal bump functions controlling curvature and its derivatives. Every quotient isometry must send an entire labelled cluster to another. After identifying the marked disks with the original one, an isometry fixes its three noncollinear points and is therefore the identity locally. If $p_i:M\to U_i\backslash M$ are free finite quotients, any quotient isometry $F$ consequently makes $F\circ p_1=p_2\circ t$ on an open disk for some element $t$ of the original group. Two local isometries agreeing on an open set agree on the connected surface, by propagation along geodesics. The equality therefore holds globally and implies $t{U}_1{t}^{-1}\subseteq U_2$, hence equality for equal subgroup orders. Nonconjugate equal-order subgroups thus have nonisometric quotients for this marked invariant metric; constant curvature is not required.