Recently Borovik and Cherlin initiated a broad study of permutation groups of finite Morley rank (fMr) where one of the main problems is to show that the only connected groups of fMr with a generically (n+2)-transitive action on a set of rank n are those of the form PGL(n+1,F). Of course the result has been known in the case of n=1 for a few decades as in this case the set is strongly minimal. In this talk, I will present results about groups acting on sets of rank 2 with a focus on those that are generically *sharply* 4-transitive. The analysis of these actions makes considerable use of the structure of groups of small rank, and as such, I will also discuss some new results on groups of rank 4.