The automorphism group of a resplendent model

To clarify notation, Aut(A) is the group of all automorphisms of A and Th(Aut(A)) is the first-order theory of that group. An L-structure A = (A, . . .), where L is some first-order language, is resplendent (consult, for example, [Hod] or [Poi]) if, whenever L ⊇ L is a language that may have some constant symbols denoting elements of A and σ is an L-sentence that has a model B such that A 4 B↾L, then A has an L-expansion A that is a model of σ. While resplendent structures are perhaps not that well known (although [Kos] is an attempt to change that), the countable ones are more so, being precisely the countable, recursively saturated structures. It should be pointed out that every countable first-order theory T having an infinite model has resplendent models of every infinite cardinality κ ≥ |T |. The impetus for proving this theorem comes from two sources. The first is the remarkable theorem of Bludov, Giraudet, Glass and Sabbagh [BGGS] that states: If T is an arbitrary first-order theory having an infinite model, then it has a model A such that Th(Aut(A)) is undecidable. In this result, the language is not restricted to being finite. Indeed, this result has the following addendum: Moreover, if κ ≥ |T | is an infinite cardinal, then A can be chosen to have cardinality κ. This theorem suggests the problem of finding a nice class of models A |= T for which Th(Aut(A)) is undecidable. The second source is the following theorem that I proved (Theorem 6.1 of [Sch]) about models of Peano Arithmetic (PA): If M is a countable, recursively saturated model of PA, then Th(Aut(M)) is undecidable. Although the proof of this result was very specific to models of PA, it, together with the theorem from [BGGS], suggests that PA