On regular surfaces of general type with numerically trivial automorphism group of order $4$

Abstract: Let $S$ be a regular minimal surface of general type over the field of complex numbers, and $\mathrm{Aut}\mathbb{Q}(S)$ the subgroup of automorphisms acting trivially on $H^*(S,\mathbb{Q})$. It has been known since twenty years that $|\mathrm{Aut}\mathbb{Q}(S)|\leq 4$ if the invariants of $S$ are sufficiently large. Under the assumption that $K_S$ is ample, we characterize the surfaces achieving the equality, showing that they are isogenous to a product of two curves, of unmixed type, and that the group $\mathrm{Aut}\mathbb{Q}(S)$ is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$. Moreover, unbounded families of surfaces with $\mathrm{Aut}\mathbb{Q}(S)\cong(\mathbb{Z}/2\mathbb{Z})^2$ are provided.