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We consider normal subgroups $N$ of the braid group $B_n$ such that thequotient $B_n/N$ is an extension of the symmetric group by an abelian group. Weshow that, if $n\geq 4$, then there are exactly 8 commensurability classes ofsuch subgroups. We define a Specht subgroup to be a subgroup of this form thatis maximal in its commensurability class. We give descriptions of the Spechtsubgroups in terms of winding numbers and in terms of infinite generating sets.The quotient of the pure braid group by a Specht subgroup is a module over thesymmetric group. We show that the modules arising this way are closely relatedto Specht modules for the partitions $(n-1,1)$ and $(n-2,2)$, working over theintegers. We compute the second cohomology of the symmetric group withcoefficients in both of these Specht modules, working over an arbitrarycommutative ring. Finally, we determine which of the extensions of thesymmetric group arising from Specht subgroups are split extensions.

Quotients of the braid group that are extensions of the symmetric group