In this paper we study the homeomorphisms of B$^{2}$ that are liftable
with respect to a simple branched covering. Since any such homeomorphism
maps the branch set of the covering onto itself and liftability is
invariant up to isotopy fi{}xing the branch set, we are dealing in
fact with liftable braids. We prove that the group of liftable braids
is fi{}nitely generated by liftable powers of half-twists around arcs
joining branch points. A set of such generators is explicitly determined
for the special case of branched coverings B$^{2}$ $\rightarrow$
B$^{2}$ . As a preliminary result we also obtain the classifi{}cation
of all the simple branched coverings of B$^{2}$.
