We compute the groups H*(Aut(F);M) and H*(Out(F);M) in a stable range, where M is obtained by applying a Schur functor to H or H, respectively the first rational homology and cohomology of F. The answer may be described in terms of stable multiplicities of irreducibles in the plethysm Sym $\circ$ Sym of symmetric powers. We also compute the stable integral cohomology groups of Aut(F) with coefficients in H or H*.