Let $X$ be a compact Riemann surface, and let $P \rightarrow X$ be a holomorphic $\mathbb P^1$-bundle over $X$. Then we know that $P$ is of form $\mathbb P(E)$ for some vector bundle $E \rightarrow X$ of rank $2$. At the end of section 5, Grothendieck proved that we have the following exact sequence: $$ 1 \rightarrow \operatorname{Aut}(E)/\mathbb{C}^* \rightarrow \operatorname{Aut}(P) \rightarrow \Gamma \rightarrow 1, $$ where $\Gamma = \{ [T] \mid [T]\text{ is isomorphism class of a line bundle }T \text{ such that } E \cong E \otimes T\}$. Hence $\Gamma$ is a finite group.

My question is *what will happen if $E$ is a stable vector bundle? Can we expect $\operatorname{Aut} (P) = e$, i.e. $\Gamma = e$?*