  We extend the periodic table of phases of free fermions in the ten-fold way
symmetry classes to a classification of free fermionic phases protected by an
arbitrary on-site unitary symmetry $\hat G$ in an arbitrary dimension. The
classification is described as a function of the real representation theory of
$\hat G$ and the data of the original periodic table. We also systematically
study in low dimensions the relationship between the free invariants and the
invariants of short-range entangled interacting phases of fermions. Namely we
determine whether a given symmetry protected phase of free fermions is
destabilized by sufficiently strong interactions or it remains stable even in
the presence of interactions. We also determine which interacting fermionic
phases cannot be realized by free fermions. Examples of both destabilized free
phases and intrinsically interacting phases are common in all dimensions.
