Extremely primitive classical groups

A primitive permutation group is said to be extremely primitive if it is not regular and a point stabilizer acts primitively on each of its orbits. By a theorem of Mann and the second and third authors, every finite extremely primitive group is either almost simple or of affine type. In this paper, we determine the examples in the case of almost simple classical groups. They comprise the 2-transitive actions of PSL(2,q) and its extensions of degree q+1, and of Sp(2m,2) of degrees 2(2m-1) +/- 2(m-1), together with the 3/2-transitive actions of PSL(2,q) on cosets of D_{q+1}, with q + 1 a Fermat prime. In addition to these three families, there are four individual examples.