Cohomologically or numerically trivial automorphisms of surfaces of general type

Our main result is the determination of the respective groups $ Aut_\mathbb{Z}(S) $ of cohomologically trivial automorphisms and $ Aut_\mathbb{Q}(S) $ of numerically trivial automorphisms for the reducible fake quadrics, that is, the surfaces $S$ isogenous to a product with $q=p_g=0$. In this way we produce new record winning examples: a surface $S$ with $|Aut_\mathbb{Q}(S)| =192$, and a surface whose cohomology has torsion with nontrivial $ Aut_\mathbb{Z}(S) \cong \mathbb{Z}/2.$