  The quantization of Class A Bianchi Type VI and VII geometries -with all six
scale factors, as well as the lapse function and the shift vector present- is
considered. A first reduction of the initial 6-dimensional configuration space
is achieved by the usage of the information furnished by the quantum form of
the linear constraints. Further reduction of the space in which the wave
function -obeying the Wheeler-DeWitt equation- lives, is accomplished by
revealing a classical integral of motion, tantamount to an extra symmetry of
the corresponding classical Hamiltonian. This symmetry generator -member of a
larger group- is linear in momenta and corresponds to G.C.T.s through the
action of the automorphism group -especially through the action of the outer
automorphism subgroup. Thus, a G.C.T. invariant wave function is found, which
depends on one combination of the two curvature invariants --which uniquely and
irreducibly characterizes the hypersurfaces t=const.
