Characterizing the existence of a Borel complete expansion

We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $\phi $ as a class of structures in a related language. We show that $\phi $ has a Borel complete expansion if and only if $S_\infty $ divides $\operatorname {Aut}(M)$ for some countable model $M\models \phi $. From this, we prove that for theories $T_h$ asserting that $\{E_n\}$ is a countable family of cross cutting equivalence relations with $h(n)$ classes, if $h(n)$ is uniformly bounded, then $T_h$ is not Borel complete, providing a converse to Theorem 2.1 of [J. Symbolic Logic 88 (2023), 418–426].