  We study quasiconformal mappings of the unit disk that have planar extension
with controlled distortion. For these mappings we prove a bound for the modulus
of continuity of the inverse map, which somewhat surprisingly is almost as good
as for global quasiconformal maps. Furthermore, we give examples which improve
the known bounds for the three point property of generalized quasidisks.
Finally, we establish optimal regularity of such maps when the image of the
unit disk has cusp type singularities.
