  We study the $H^3$ invariant of a group homomorphism $\phi:G \rightarrow
\mathrm{Out}(A)$, where $A$ is a classifiable C$^*$-algebra. We show the
existence of an obstruction to possible $H^3$ invariants arising from
considering the unitary algebraic $K_1$ group. In particular, we prove that
when $A$ is the Jiang--Su algebra $\mathcal{Z}$ this invariant must vanish. We
deduce that the unitary fusion categories $\mathrm{Hilb}(G, \omega)$ for
non-trivial $\omega \in H^3(G, \mathbb{T})$ cannot act on $\mathcal{Z}$.
