Vector bundle generated by global sections and pullback

Question

Let $\mathcal{E}$ be a locally free sheaf on a smooth projective $X$ which is generated by global sections. I.e. there is a surjection $H^0(\mathcal{E})\otimes \mathcal{O}_X\twoheadrightarrow \mathcal{E}$. If $f$ is an automorphism acting on $X$, then we have a natural isomorphism of vector spaces $H^0(\mathcal{E})\xrightarrow{\sim} H^0(f^*\mathcal{E})$. So I think its not true that for a general coherent sheaf $\mathcal{E}$ such an isomorphism induces a map $\mathcal{E} \rightarrow f^*\mathcal{E}$, not even when its generated by global sections. But my question is: Does the isomorphism $H^0(\mathcal{E}) \xrightarrow{\sim} H^0(f^*\mathcal{E})$ for $\mathcal{E}$ globally generated and locally free induce an iso $\mathcal{E} \rightarrow f^*\mathcal{E}$?
Idea: We have an exact sequence $0 \rightarrow K\rightarrow H^0(\mathcal{E})\otimes \mathcal{O}\rightarrow \mathcal{E} \rightarrow 0$.
Now $K$ is also locally free and I think it's also globally generated, i.e. can be written as $K=V\otimes \mathcal{O}$ for $V \subset H^0(\mathcal{E})$. Then I can I apply $f^*$ to the middle and left term to get another exact sequence and a commutative diagram of short exact sequences with a map of co-kernels $\mathcal{E} \rightarrow f^*\mathcal{E}$…?

Answer 1

A simple counterexample is when $X = \mathbb{P}^1 \times \mathbb{P}^1$, $f$ swaps the factors, and $\mathcal{E} \cong \mathcal{O}(1,0)$.