  We search for a real bosonic and fermionic action in four dimensions which
both remain invariant under local Weyl transformations in the presence of
non-metricity and contortion tensor. In the presence of the non-metricity
tensor the investigation is extended to Weyl $(W_n, g)$ space-time while when
the torsion is encountered we are restricted to the Riemann-Cartan $(U_4, g)$
space-time. Our results hold for a subgroup of the Weyl-Cartan $(Y_4, g)$
space-time and we also calculate extra contributions to the conformal gravity.
