For any group G and an abelian group A, the second cohomology of G with coefficients in A is in one-to-one correspondence with group extensions of G by A. In 2018 Brendle and Margalit showed there is a non-split extension of the symmetric group by the level 4 braid group. In this talk I will begin by discussing the connection between subgroups of the braid group and extensions of the symmetric group. Then I will construct the 2–cocylce corresponding to the extension of the symmetric group by abelianization of the pure braid group and show this is an element of order 2.