  We compute the groups $H^*(\mathrm{Aut}(F_n); M)$ and $H^*(\mathrm{Out}(F_n);
M)$ in a stable range, where $M$ is obtained by applying a Schur functor to
$H_\mathbb{Q}$ or $H^*_\mathbb{Q}$, respectively the first rational homology
and cohomology of $F_n$. For reasons which are not conceptually clear, taking
coefficients in $H_\mathbb{Q}$ and its related modules behaves in a far less
trivial way than taking coefficients in $H^*_\mathbb{Q}$ and its related
modules. The answer may be described in terms of stable multiplicities of
irreducibles in the plethysm $\mathrm{Sym}^k \circ \mathrm{Sym}^l$ of symmetric
powers.
  We also compute the stable integral cohomology groups of $\mathrm{Aut}(F_n)$
with coefficients in $H$ or $H^*$, respectively the first integral homology and
cohomology of $F_n$, and compute the stable cohomology with coefficients in
Schur functors of $H$ or $H^*$ modulo small primes.
