We characterize the cyclic branched covers of the 2-sphere where every homeomorphism of the sphere lifts to a homeomorphism of the covering surface. This answers the question that appeared in an early version of the erratum of Birman and Hilden .
"Lifting Homeomorphisms and Cyclic Branched Covers of Spheres." Michigan Math. J. 66 (4) 885 - 890, November 2017. https://doi.org/10.1307/mmj/1508810819