  For i = 1,2, let Gamma_i be a lattice in a simply connected, solvable Lie
group G_i, and let X_i be a connected Lie subgroup of G_i. The double cosets
Gamma_igX_i provide a foliation F_i of the homogeneous space Gamma_i\G_i. Let f
be a continuous map from Gamma_1\G_1 to Gamma_2\G_2 whose restriction to each
leaf of F_1 is a covering map onto a leaf of F_2. If we assume that F_1 has a
dense leaf, and make certain technical technical assumptions on the lattices
Gamma_1 and Gamma_2, then we show that f must be a composition of maps of two
basic types: a homeomorphism of Gamma_1\M_1 that takes each leaf of F_1 to
itself, and a map that results from twisting an affine map by a homomorphism
into a compact group.
  We also prove a similar result for many cases where G_1 and G_2 are neither
solvable nor semisimple.
