Automorphisms of Extensions of Q by a Direct Sum of Finitely Many Copies of Q

Let G be an extension of . by a direct sum of r copies of .. (1) If G is abelian, then G is a direct sum of r + 1 copies of . and AutG . GL(r + 1, Q); (2) If G is non-abelian, then G is a direct product of an extraspecial .-group E and m copies of ., where E/.E is a linear space over Q with dimension 2n and m + 2n = r. Furthermore, let AutG.G be the normal subgroup of AutG consisting of all elements of AutG which act trivially on the derived subgroup G. of G, and AutG/.G,.GG be the normal subgroup of AutG consisting of all central automorphisms of G which also act trivially on the center .G of G. Then (i) The extension 1 . AutG.G . AutG . AutG. . 1 is split; (ii) AutG.G/AutG/.G,.GG . Sp(2n,Q) . (GL(m, Q) . .(m)); (iii) AutG/.G,.GG/InnG . .(2nm).