For equivalence of unbranched coverings of topological spaces, there is a criteria:

Two coverings (unbranched) $p_1\colon Y_1\rightarrow X$ and $p_2\colon Y_2\rightarrow X$ are equivalent iff for some $q\in X$ and $\bar{q_1}\in p_1^{-1}(q)$ and $\bar{q_2}\in p_2^{-1}(q)$, the induced subgroups $p_*\pi_1(Y_1,\bar{q_1})$ and $p_*\pi_1(Y_2,\bar{q_2})$ are conjugate in $\pi_1(X,q)$.

Is there any criteria (particularly group theoretic) for equivalence of branched coverings of Riemann surfaces?