Extremely primitive sporadic and alternating groups

A non-regular primitive permutation group is said to be extremely primitive if 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 a recent paper, we classified the extremely primitive almost simple classical groups, and in this note we determine the examples with a sporadic or alternating socle. We obtain two infinite families for A_n (or S_n); they comprise the natural 2-primitive action of n points, plus the action on partitions of {1,...,n} into subsets of size n/2 (with n/2 odd). There are twenty examples for sporadic groups, including the rank 6 representation of Co_2 on the cosets of McL.