  We introduce a way of describing cohomology of the symmetric groups with
coefficients in Specht modules over Z or F_p. We study i-th-degree cohomology
for i in {0,1,2}. The focus lies on the isomorphism type of second-degree
cohomology of integral Specht modules. Unfortunately, only in few cases can we
determine this exactly. In many cases we obtain only some information about the
prime divisors of the cohomology group's order. The most important tools we use
are the Zassenhaus algorithm, the Branching Rules, Bockstein type
homomorphisms, and the results from [Burichenko et al., 1996].
