Lifting of homeomorphisms to 4-sheeted branched coverings of a disk

Let p : X → D be a simple, possibly not connected, 4-sheeted branched covering of a closed 2-dimensional disk D with n branch values A 1 , … , A n . The isotopy classes of homeomorphisms of D which are fixed on the boundary of D and permute the branch values form a braid group B n . Some of these homeomorphisms can be lifted to homeomorphisms of X. They form a subgroup L ( p ) of finite index in B n . For each equivalence class of coverings we find a set of generators for L ( p ) which contains between n and n + 4 elements, depending on the equivalence class of the covering, and the generators are powers of half-twists.