Limit Groups and Automorphisms of κ-Existentially Closed Groups

The structure of automorphism groups of κ-existentially closed groups are studied by Kaya-Kuzucuo{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and |Aut(G)|=2^κ whenever κ is an inaccessible cardinal and G is the unique κ-existentially closed group of cardinality κ. The cardinality of the automorphism group of a κ-existentially closed group of cardinality λ>κ is asked in Kourovka Notebook Question 20.40. Here we answer positively the promised case κ=λ namely: If G is a κ-existentially closed group of cardinality κ, then |Aut(G)|=2κ. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal κ, there exist universal groups of cardinality κ.