  A formula is presented for the modular transformation matrix S for any simple
current extension of the chiral algebra of a conformal field theory. This
provides in particular an algorithm for resolving arbitrary simple current
fixed points, in such a way that the matrix S we obtain is unitary and
symmetric and furnishes a modular group representation. The formalism works in
principle for any conformal field theory. A crucial ingredient is a set of
matrices S^J_{ab}, where J is a simple current and a and b are fixed points of
J. We expect that these input matrices realize the modular group for the torus
one-point functions of the simple currents. In the case of WZW-models these
matrices can be identified with the S-matrices of the orbit Lie algebras that
we introduced in a previous paper. As a special case of our conjecture we
obtain the modular matrix S for WZW-theories based on group manifolds that are
not simply connected, as well as for most coset models.
