Saturated models

The automorphism group

In a recent paper Nurkhaidarov and Schmerl 2011 proved the following: Let $\kappa$ be regular, uncounale, and such that $\kappa^{<\kappa}=\kappa$. For each completion $T\supseteq PA$ let $M^\kappa_T$ be the saturated model of $T$ of cardinality $\kappa$. There is a set $\mathcal T$ of completions of $PA$, such that $|{\mathcal T}|=2^{\aleph_0}$ and for all $T, T'\in {\mathcal T}$, if $T\not=T'$, then $\aut(M^\kappa_T)\not\cong {\rm Aut}(M^\kappa_{T'})$. The following question is left open: Are there $T$ and $T'$ such that $T\not=T'$ and $\aut(M^\kappa_T)\cong {\rm Aut}(M^\kappa_{T'})$?