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 A1,…,An. The isotopy classes of homeomorphisms of D which are fixed on the boundary of D and permute the branch values form a braid group Bn. Some of these homeomorphisms can be lifted to homeomorphisms of X. They form a subgroup L(p) of finite index in Bn. 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.