Second cohomology groups and finite covers of infinite symmetric groups

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