  In this paper we analyze the structure of transitive permutation groups that
have trivial four point stabilizers, but some nontrivial three point
stabilizer. In particular we give a complete, detailed classification when the
group is simple or quasisimple. This paper is motivated by questions concerning
the relationship between fixed points of automorphisms of Riemann surfaces and
Weierstrass points and is a continuation of the authors' earlier work.
