Abstract. When studying automorphism groups, one is interested to what extent a model is recoverable from its automorphism group. I will show that if M is a countable arithmetically saturated model of PA then the automorphism group of M can recognize if a maximal subgroup is a stabilizer of an element a, such that a is greater than any standard and smaller than any non-standard definable element. I will also show existence of countable arithmetically saturated models of False Arithmetic with the same standard system such that their automorphism groups are not isomorphic.