Characterizing covers via simple closed curves

Given two finite covers $$p: X \rightarrow S$$ and $$q: Y of a connected, oriented, closed surface S genus at least 2, we attempt to characterize the equivalence p q in terms which curves lift simple curves. Using Teichmüller theory complex curves, show that regular are equivalent if for any curve $$\gamma \subset , $$ lifts on only it does Y. When abelian, also give characterization powers