Limit Groups and Automorphisms of $\kappa$-Existentially Closed Groups

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