Automorphism group of ruled surface

Question

Let $C$ be an elliptic curve over the complex numbers. Consider a nontrivial extension $$ 0 \to \mathcal O_C \to E \to \mathcal O_C \to 0 $$ of rank 2 of the structure sheaf of $C$. This defines a ruled surface $X = \mathbb P(E)$ over $C$.

Is the automorphism group of $X$ transitive?

I ask because I'm looking for a compact Kahler manifold $X$ with nef $-K_X$ whose automorphism group is not transitive. The Albanese variety of $X$ should also be nontrivial, ruling out the obvious Fano candidates. The ruled surface above has nef $-K_X$ and its Albanese variety is $C$ itself, so it almost fits the bill.

Answer 1

Any automorphisms of $X$ lies over an automorphism of $C$. It seems to me that there is a unique section $C \to X$ with trivial normal bundle, so this section should be carried to itself by an automorphism, which seems to show that the automorphism group is not transitive.