  We study metric teleparallel geometries, which can either be defined through
a Lorentzian metric and flat, metric-compatible affine connection, or a tetrad
and a flat spin connection, which are invariant under the transitive action of
a four-dimensional Lie group on their spatial equal-time hypersurfaces. There
are three such group actions, and their corresponding spatial hypersurfaces
belong to the Bianchi types II, III and IX, respectively. For each of these
three symmetry groups, we determine the most general teleparallel geometry, and
find that it is parametrized by six functions of time, one of which can be
eliminated by the choice of the time coordinate. We further show that these
geometries are unique up to global Lorentz transformations, coordinate
transformations and changes of the choice of the parameter functions.
