Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some intermediate covering is already a manifold, and I am looking for a method to detect such cases.
More precisely, say you are given a subgroup $G \leq \pi_1 X$. Is there a criterion to check if the induced covering orbifold has a trivial orbifold structure?