Second cohomology groups and finite covers of infinite symmetric groups

For Ω an infinite set, k⩾2 and W the set of k-sets from Ω, there is a natural closed permutation group Γk which is a non-split extension . We classify the closed subgroups of Γk which project onto Sym(Ω). The question arises in model theory as a problem about finite covers, but here we formulate and solve it in algebraic terms.