Axiomatizability of generic automorphisms

Given an $L$-theory $T$, we define the language $L_{\sigma}$ to be the language obtained by adding a new unary function symbol $\sigma$ to the language and we define $T_{\sigma}$ to be the $L_{\sigma}$-theory which extends $T$ with axioms asserting that $\sigma$ is an automorphism. When $T$ is a model complete theory, we say $T$ admits the axiomatizability of a generic automorphism if $T_{\sigma}$ has a model companion.

Kikyo showed that if $T$ is unstable and models of $T_{\sigma}$ have the amalgamation property then $T_{\sigma}$ does not have a model companion [Kik14]. The same conclusion was proved by Kikyo and Shelah for theories $T$ with the strict order property [KS02].