Limit groups and automorphisms of κ-existentially closed groups

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