  Symmetry assumptions on the geometrical framework have provided successful
mechanisms to develop physically meaningful solutions to many problems. In
tele-parallel gravity, invariance of the frame and spin-connection under a
group of motions defines an affine symmetry group. Here, we assume there exists
a three-dimensional group of affine symmetries acting simply transitively on a
spatial hypersurface and that this group of symmetry actions defines our affine
frame symmetry group. We determine the general form of the co-frame and spin
connection for each spatially homogeneous Bianchi type. We then construct the
corresponding field equations for $f(T)$ tele-parallel gravity. We show that if
the symmetry group is of Bianchi type A ($I$, $II$, $VI_0$, $VII_0$, $VIII$ or
$IX$) then there exists a co-frame/spin connection pair that is consistent with
the antisymmetric part of the field equations of $f(T)$ tele-parallel gravity.
For those geometries having a Bianchi type B symmetry group ($IV$, $V$, $VI_h$,
$VII_h$), we find that in general these geometries are inconsistent with the
antisymmetric part of the $f(T)$ tele-parallel gravity field equations unless
the theory reduces to an analog of General Relativity with a cosmological
constant.
