Systems of generators of automaton permutations groups

Systems of induced generating actions of automaton permutations groups on words of length r are investigated. A family of irreducible systems of generators is constructed, and the cardinality of such systems is found to be related with r. The infinite generativity (even in the topological sense) of groups of all automaton permutations, finite automaton permutations, and finitary automaton permutations is established; the well-known proof of this fact contained an error.