Automorphisms of $$\kappa $$-existentially closed groups

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We investigate the automorphisms of some \(\kappa \)-existentially closed groups. In particular, we prove that Aut(G) is the union of subgroups of level preserving automorphisms and \(|Aut(G)|=2^{\kappa }\) whenever \(\kappa \) is inaccessible and G is the unique \(\kappa \)-existentially closed group of cardinality \(\kappa \). Indeed, the latter result is a byproduct of an argument showing that, for any uncountable \(\kappa \) and any group G that is the limit of regular representation of length \(\kappa \) with countable base, we have \(|Aut(G)|=\beth _{\kappa +1}\), where \(\beth \) is the beth function. Such groups are also \(\kappa \)-existentially closed if \(\kappa \) is regular. Both results are obtained by an analysis and classification of level preserving automorphisms of such groups.