  Let F_{n+m} be the free group of rank n+m, with generators x_1,...,x_{n+m}.
An automorphism \phi of F_{n+m} is called partially symmetric if for each 1 \le
i \le m, \phi(x_i) is conjugate to x_j or x_j^{-1} for some 1 \le j \le m. Let
\Sigma\Aut_n^m be the group of partially symmetric automorphisms. We prove that
for any m \ge 0 the inclusion \Sigma\Aut_n^m \to \Sigma\Aut_{n+1}^m induces an
isomorphism in rational homology for dimensions i satisfying n \ge
(3(i+1)+m)/2, with a similar statement for the groups P\Sigma\Aut_n^m of pure
partially symmetric automorphisms. We also prove that for any n \ge 0 the
inclusion \Sigma\Aut_n^m \to \Sigma\Aut_n^{m+1} induces an isomorphism in
rational homology for dimensions i satisfying m > (3i-1)/2.
