Anomalous symmetries of classifiable C*-algebras

We study the $H^3$ invariant of a group homomorphism $\phi:G \rightarrow $\mathrm{O}\mathrm{u}\mathrm{t}$(A)$,where$A$isaclassifiableC$^*$-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 $\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}(G,\omega )$ for non-trivial $\omega \in H^3(G,\mathbb{T})$ cannot act on $\mathcal{Z}$.