Base sizes for finite unitary and symplectic groups with solvable stabilisers

Let G be a transitive permutation group on a finite set with solvable point stabiliser and assume that the solvable radical of G is trivial. In 2010, Vdovin conjectured that the base size of G is at most 5. Burness proved this conjecture for primitive G. The problem was reduced by Vdovin in 2012 to the case when G is an almost simple group, and reduced to groups of Lie type by Baykalov and Burness. This is the second paper of the series devoted to the study of Vdovin's conjecture for classical groups. In the first paper, we prove a strong form of the conjecture for almost simple groups with socle isomorphic to PSL_n(q). In this paper, we extend this result to almost simple groups with socle isomorphic to PSU_n(q) and PSp_n(q). The final paper will establish the conjecture for orthogonal groups. Together, these three paper will complete the proof of Vdovin's conjecture for all almost simple classical groups.