  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. From this, we show that $\Phi$ has a Borel complete expansion
if and only if $S_\infty$ divides $Aut(M)$ for some countable model $M\models
\Phi$. Using 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 \cite{LU}.
