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 $$\operatorname{Aut}^\circ(P)\subset G\subset \operatorname{Bir}( X\times {\mathbb P}^1).$$ We get a quite complete answer if $\operatorname{Aut}^\circ(X)$ is a maximal, connected, algebraic subgroup of $\operatorname{Bir}(X)$, and $P$ is sufficiently unstable. This gives examples of connected, algebraic subgroups of $\operatorname{Bir}({\mathbb P}^4)$ that are not contained in any maximal one. Version 2: references updated.