  We establish a Tukia-type theorem for uniform quasiconformal groups of a
Carnot group. More generally we establish a fiber bundle version (or foliated
version) of Tukia theorem for uniform quasiconformal groups of a nilpotent Lie
group whose Lie algebra admits a diagonalizable derivation with positive
eigenvalues. These results have applications to quasi-isometric rigidity of
solvable groups [DFX].
