  We study some particular modifications of gravity in search for a natural way
to unify the gravitational and electromagnetic interaction. The certain
components of connection in the appearing variants of the theory can be
identified with electromagnetic potential. The methods of adding matter in the
form of scalar and spinor fields are studied. In particular, the expansion of
the local symmetry group up to $GL(2,C)$ is explored, in which equations of
Einstein, Maxwell and Dirac are reproduced for the theory with Weyl spinor.
