Cohomologically or numerically trivial automorphisms of surfaces of general type

Our main result is the determination of the respective groups $Au{t}_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms and $Au{t}_{\mathbb{Q}}(S)$ 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 $\mathrm{\mid }Au{t}_{\mathbb{Q}}(S)\mathrm{\mid }=192$, and a surface whose cohomology has torsion with nontrivial $Au{t}_{\mathbb{Z}}(S)\cong \mathbb{Z}\mathrm{/}2.$