  Keisler proved that if $\theta$ is a strong limit cardinal and $\lambda$ is a
singular cardinal, then the transfer relation $\theta\longrightarrow\lambda$
holds. In a previous paper, we studied initial elementary submodels of the
$\lambda$-like models produced in the proof of Keisler's transfer theorem when
$\theta$ is further assumed to be regular i.e., $\theta$ is strongly
inaccessible. In this paper we deal with a much more difficult situation. Some
years ago Ali Enayat asked the author whether Keisler's singular-like models
can have elementary end extensions. We give a positive answer to this question.
