The feedback operation and a class of group automata

We study the behavior of the inner group of an automaton under the feedback operation if no application of this operation takes the automaton out of the class of group automata. The main result is that if the inner group of an automaton is not a cyclic group of prime order, then there exists an automaton for this group such that the application of the feedback operation a sufficient number of times yields an automaton with an arbitrary subgroup of the divisible symmetric group for that number of states. Together with this, we study the change of the group of an automaton under a single application of the feedback operation.