  We give a description of the group of all quasisymmetric self-maps of the
Julia set of $f(z)=z^2-1$ that have orientation preserving homeomorphic
extensions to the whole plane. More precisely, we prove that this group is the
uniform closure of the group generated by the Thompson group of the unit circle
and an inversion. Moreover, this result is quantitative in the sense that
distortions of the approximating maps are uniformly controlled by the
distortion of the given map.
