  The problems encountered in trying to quantize the various cosmological
models, are brought forward by means of a concrete example. The Automorphism
groups are revealed as the key element through which G.C.T.'s can be used for a
general treatment of these problems. At the classical level, the time dependent
automorphisms lead to significant simplifications of the line element for the
generic spatially homogeneous geometry, without loss of generality. At the
quantum level, the ''frozen'' automorphisms entail an important reduction of
the configuration space --spanned by the 6 components of the scale factor
matrix-- on which the Wheeler-DeWitt equation, is to be based. In this spirit
the canonical quantization of the most general minisuperspace actions --i.e.
with all six scale factor as well as the lapse function and the shift vector
present-- describing the vacuum type II, I geometries, is considered. The
reduction to the corresponding physical degrees of freedom is achieved through
the usage of the linear constraints as well as the quantum version of the
entire set of all classical integrals of motion.
