Automorphism groups of $\mathbb{P}^1$-bundles over ruled surfaces

Abstract: We classify the pairs $(X,\pi)$, where $\pi\colon X\to S$ is a $\mathbb{P}^1$-bundle over a non-rational ruled surface $S$ and $\mathrm{Aut}^\circ(X)$ is relatively maximal, i.e., maximal with respect to the inclusion in the group $\mathrm{Bir}(X/S)$. The results hold over any algebraically closed field of characteristic zero.