  A map on finitely many fermionic modes represents a unitary evolution if and
only if it preserves canonical anti-commutation relations. We use this
condition for the classification of fermionic cellu- lar automata (FCA) on
Cayley graphs of finite groups in two simple but paradigmatic case studies. The
physical properties of the solutions are discussed. Finally, features of the
solutions that can be extended to the case of cellular automata on infinite
graphs are analyzed.
