The automorphism tower of a centerless group without Choice

For a centerless group G, we can define its automorphism tower. We define G^α: G⁰ = G, G^α+1 = Aut(G^α) and for limit ordinals G^δ = ∪_α>δ. Let τ_G be the ordinal when the sequence stabilizes. Thomas' celebrated theorem says τ_G>(2^|G|)⁺and more. If we consider Thomas' proof too set theoretical (using Fodor's lemma), we have here a more direct proof with little set theory. However, set theoretically we get a parallel theorem without the Axiom of Choice. Moreover, we give a descriptive set theoretic approach for calculating an upper bound for τ_G for all countable groups G (better than the one an analysis of Thomas' proof gives). We attach to every element in G^α, the αth member of the automorphism tower of G, a unique quantifier free type over G (which is a set of words from G *). This situation is generalized by defining "(G, A) is a special pair".