We investigate gauge anomalies in the context of orbifold conformal field theories. Such anomalies manifest as failures of modular invariance in the constituents of the orbifold partition function. We review how this irregularity is classified by cohomology and how extending the orbifold group can remove it. Working with such extensions requires an understanding of the consistent ways in which extending groups can act on the twisted states of the original symmetry, which leads us to a discrete torsionlike choice that exists in orbifolds with trivially acting subgroups. We review a general method for constructing such extensions and investigate its application to orbifolds. Through numerous explicit examples we test the conjecture that consistent extensions should be equivalent to (in general multiple copies of) orbifolds by nonanomalous subgroups.