Rational homological stability for groups of symmetric automorphisms of free groups

Let F_n be the free group of rank n, with generating set S={x_1,…,x_n}. An automorphism \phi of F_n is called symmetric if for each 1\leq i\leq n, \phi(x_i) is conjugate to x_j or x_j^{-1} for some 1\leq j\leq n. Let \Sigma Aut(F_n) be the group of symmetric automorphisms. We prove that the inclusion \Sigma Aut(F_n) \rightarrow \Sigma Aut(F_{n+1}) induces an isomorphism in rational homology for n>(3i-1)/2.