Equations of $G$-minimal conic bundles with degree $4$

The first nontrivial case of relatively $G$-minimal conic bundles being $G$-minimal is considered. The number $r$ of singular fibers equals $4$. Classification gives explicit equations of minimal conic bundles $(S,G)$ and an explicit action of the group $G$ on the Picard group $\operatorname{Pic}(S)$ and on the surface $S$.