κ-Existentially closed groups and centralizers

Let κ be an infinite cardinal. In 2018, Kegel and Kuzucuoğlu proved that, if κ>ℵ0, any two κ-existentially closed groups of cardinality κ are isomorphic. As a first result, we show that if we allow the cardinality of the group to be greater than κ, then the above uniqueness is no longer true. In fact, one can get as many pairwise non-isomorphic κ-existentially closed groups of cardinality κ+ as possible, provided that κ is less than the first strongly inaccessible cardinal. Secondly, we address the following question inspired by Hickin and Macintyre: Given a κ-existentially closed group G of cardinality κ for some infinite cardinal κ and a subgroup A generated by fewer than κ elements, when is CG(A)/Z(A) isomorphic to G? We prove that this holds exactly when Z(A)=1.