An abstract smooth surface is a two-dimensional smooth manifold. Thus it is a Hausdorff second-countable space with an atlas of plane-valued coordinate charts whose transition maps are smooth.
A finite group acting freely by diffeomorphisms on a smooth manifold has a smooth quotient. Choose each coordinate neighbourhood disjoint from all its nontrivial translates and transport its chart through the quotient map. The resulting transition maps are restrictions of the original transition maps composed with group elements.

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