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

We classify the pairs $(X,\pi )$, where $\pi \text{\hspace{0.17em}⁣}:X\to S$ is a ${\mathbb{P}}^1$-bundle over a non-rational geometrically ruled surface $S$ and ${\mathrm{A}\mathrm{u}\mathrm{t}}^{\circ }(X)$ is relatively maximal, i.e., maximal with respect to the inclusion in the group $\mathrm{B}\mathrm{i}\mathrm{r}(X\mathrm{/}S)$. The results hold over any algebraically closed field of characteristic zero.