In ``Unitary equivalence of automorphisms of separable $C^*$ algebras,'' Lupini proves non-classifiability and dichotomy theorems of considerable generality, with in fact just one class of exceptions: the nonunital continuous-trace $C^*$ algebras. We discuss recent joint work with Lupini and Panagiotopoulos showing that the theorems in question do, in fact, break down on this class. This entails analysis of cohomological invariants attaching to automorphisms of algebras in this class.