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

Let $S$ be a regular minimal surface of general type over the field of complex numbers, and ${\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathbb{Q}}(S)$ the subgroup of automorphisms acting trivially on $H^{\ast }(S,\mathbb{Q})$. It has been known since twenty years that $\mathrm{\mid }{\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathbb{Q}}(S)\mathrm{\mid }\le 4$ 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 ${\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathbb{Q}}(S)$ is isomorphic to $(\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^2$. Moreover, unbounded families of surfaces with ${\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathbb{Q}}(S)\cong (\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^2$ are provided.