Automorphisms of $\mathbb{F}_{n}$.

Question

Let $\mathbb{F}_n =\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}(n)\oplus\mathcal{O})$ with $n\geq1$. I was trying to calculate $Aut(\mathbb{F}_{n}) = G$. It is not hard to see that an automorphism maps fibres (of the $\mathbb{P}^1$-bundle) to fibres so there is a morphism $G \rightarrow Aut(\mathbb{P}^1) = PGl_2$. My question is about the kernel, I have read that this semi-direct product of $\mathbb{C}^*$ with $H^{0}(\mathbb{P}^1, \mathcal{O}(n))$. This doesn't look so random since $\mathbb{F}_n = \mathbb{P}(\mathcal{O}(n)\oplus \mathcal{O})$, I can see how $ \mathbb{C}^*$ acts by multiplication on the trivial factor, but how does the semi-direct product act?