  In this paper, we prove that any group of diffeomorphisms acting on the
2-sphere and properly extending the conformal group of M\"obius transformations
must be at least 4-transitive or, more precisely, arc 4-transitive. In
addition, we show that any such group must always contain an element of
positive topological entropy. We also provide an elementary characterization,
in terms of transitivity, of the M\"obius transformations within the full group
of sphere diffeomorphisms.
