Jónsson groups of various cardinalities

A group $G$ is Jónsson if $|H| < |G|$ whenever $H$ is a proper subgroup of $G$. Using an embedding theorem of Obraztsov it is shown that there exists a Jónsson group $G$ of infinite cardinality $κ$ if and only if there exists a Jónsson algebra of cardinality $κ$. Thus the question as to which cardin