On the homomorphisms of strongly connected finite automata into substitution automata

An approach to a description of homomorphisms of strongly connected finite automata into substitution automata is suggested. This approach is based on special homomorphisms such that their composition with homomorphism of an automaton considered into substitution automaton preserves the group of an image. By means of this approach for strongly connected automata a criterion is given for the existence of nontrivial substitutional homomorphic image wish commutative partial transition functions.