Cohomology of automorphism groups of free groups with twisted coefficients

We compute the groups $H^{\ast }(\mathrm{A}\mathrm{u}\mathrm{t}(F_n);M)$ and $H^($\mathrm{O}\mathrm{u}\mathrm{t}$(F_n); M)$inastablerange,where$M$isobtainedbyapplyingaSchurfunctorto$H_$\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 $H_{\mathbb{Q}}^{\ast }$ and its related modules. The answer may be described in terms of stable multiplicities of irreducibles in the plethysm ${\mathrm{S}\mathrm{y}\mathrm{m}}^k\circ {\mathrm{S}\mathrm{y}\mathrm{m}}^l$ of symmetric powers. We also compute the stable integral cohomology groups of $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ with coefficients in $H$ or $H^{\ast }$, respectively the first integral homology and cohomology of $F_n$, and compute the stable cohomology with coefficients in Schur functors of $H$ or $H^{\ast }$ modulo small primes.