We present an algorithm to computeH2(G,U) for a finite groupGand finite abelian groupU(trivialG-module). The algorithm returns a generating set for the second cohomology group in terms of representative 2-cocycles, which are given explicitly. This information may be used to find presentations for corresponding central extensions ofUbyG. An application of the algorithm to the construction of relative (4t, 2,4t, 2t) -difference sets is given.