We determine all firm and residually connected rank 2 geometries on which PSL(2, q) acts flag-transitively, residually weakly primitively and locally two-transitively, in which one of the maximal parabolic subgroups is isomorphic to A 4 , S 4 , A 5 , PSL(2, q) or PGL(2, q), where q divides q, for some prime-power q.