Automorphisms of unstable ${\mathbb{P}}^1$-bundles

Let $P\to X$ be a ${\mathbb{P}}^1$-bundle over a variety $X$. The aim of this note is to understand all connected, algebraic groups $${\mathrm{Aut}}^{\circ }(P)\subset G\subset \mathrm{Bir}(X\times {\mathbb{P}}^1)\mathrm{.}$$ We get a quite complete answer if ${\mathrm{Aut}}^{\circ }(X)$ is a maximal, connected, algebraic subgroup of $\mathrm{Bir}(X)$, and $P$ is sufficiently unstable. This gives examples of connected, algebraic subgroups of $\mathrm{Bir}({\mathbb{P}}^4)$ that are not contained in any maximal one. Version 2: references updated.