Automorphisms of fiber surfaces of genus $2$, inducing the identity in cohomology

Let $S$ be a complex non-singular projective surface of general type with a genus $2$ fibration and $\chi (\mathcal O_S)\geq 5$. Let $G \subset\operatorname{Aut}S$ be a non-trivial subgroup of automorphisms of $S$, inducing trivial actions on $H^i(S,\mathbb{Q})$ for all $i$. Then $\vert G\vert=2$, $K_S^2=4\chi (\mathcal O_S)$ and $q(S)=1$. Examples of such surfaces are given.