Pullbacks and isomorphisms of vector bundles

It is usually the case that if you have a nontrivial automorphism $g : X \to X$ of a complex manifold then $g^* E$ is not isomorphic to $E$. It is easiest to give an example for holomorphic line bundles.

Let $X$ be $\mathbb{P}^1 \times \mathbb{P}^1$ and $E = \mathcal{O}(1,0)$ the sheaf associated to the ``horizontal'' divisor. Then the automorphism $g : X \to X$ swapping the two copies of $\mathbb{P}^1$ then $g^* \mathcal{O}(1,0) = \mathcal{O}(0,1)$ which are not isomorphic (think about linear equivalences of divisors).

In general, for holomorphic line bundles, it is usually easy to detect when this occurs because $g$ acts quite explicitly on $\mathrm{Pic}(X)$ in terms of moving around divisors (e.g. $g$ takes ''horizontal lines'' to ''vertical lines'' in the previous example).

For complex (smooth) line bundles, there is a similar story. The first Chern class $c_1(L) \in H^2(X, \mathbb{Z})$ is a complete invariant so you just need to see how an automorphism acts on $H^2(X, \mathbb{Z})$. In the previous case, $H^2(X, \mathbb{Z}) = \mathbb{Z}^{\oplus 2}$ and $g$ swaps the factors of $\mathbb{Z}$ so we see that $g^* E$ and $E$ are not isomorphic as complex line bundles.

Finally, for real line bundles, the first Steifel-Whitney class $w_2(L) \in H^1(X, \mathbb{Z}/2\mathbb{Z})$ is a complete invariant so you just need to see how an automorphism acts on $H^1$. For example, let $X$ be the complex torus defined by the lattice $\Lambda = \mathbb{Z} \oplus \mathbb{Z}i$ and consider $g$ to be multiplication by $i$ (i.e. an elliptic curve with CM). Then, $$ H^1(X, \mathbb{Z}/2\mathbb{Z}) = (\mathbb{Z}/2\mathbb{Z})^{\oplus 2} $$ and $g$ should act on $H^1$ by swapping the copies of $\mathbb{Z}/2\mathbb{Z}$.