Automorphisms of $\mathbb{P}^1$-bundles over rational surfaces in positive characteristic

We study $\mathbb{P}^1$-bundles over smooth projective rational surfaces over an algebraically closed field of characteristic $p>0$, together with the actions of their neutral automorphism groups. After an equivariant square birational reduction to the projective plane or a Hirzebruch surface, we classify the possible induced actions on the base and identify the families relevant to relative maximality. For $p\geq 3$, we prove that every such $\mathbb{P}^1$-bundle is equivariantly square birational to one whose neutral automorphism group is relatively maximal, and we classify all bundles with this property. We also describe all equivariant square birational maps between these relatively maximal bundles and determine which are stiff or superstiff, the analogues of rigidity and superrigidity for $\mathbb{P}^1$-bundles.