Pullbacks and isomorphisms of vector bundles I want to clarify certain details regarding pullbacks of vector bundles. I suspect my question will be equivalent in every possible setting: topological, differentiable and holomorphic vector bundles.
Setting
Let $X$ be a manifold and $E$ a vector bundle over $X$. We consider automorphisms of $X$, namely those isomorphisms in the appropriate category (homeomorphisms, diffeomorphisms, biholomorphisms...). Let $g:X\rightarrow X$ be one such automorphism. I can construct the pullback bundle by either declaring transition functions from those of $E$
$$
E \equiv \{U_i, \psi_{ij}\} \implies g^*E \equiv \{g^{-1}(U_i), \psi_{ij}\circ g\}
$$
or even more explicitly by
$$
g^*E = \{ (x,e)\in X\times E| g(x)=\pi(e)\}
$$
I already know that there is a commutative diagram involving these bundles
$$\begin{array}
*g^*E & \stackrel{\hat{g}}{\longrightarrow} & E \\
\downarrow{p_1} & & \downarrow{\pi} \\
X & \stackrel{g}{\longrightarrow} & X  
\end{array}
$$
and $g(x,e)=e$ covers the automorphism $g$.
Question
I want to clarify if it is possible that $g^*E$ is isomorphic to the original bundle, meaning there is a commutative diagram as above covering the identity on X. I know this is a stronger condition, but as I have $g$ an automorphism of $X$, I suspect this could be possible in some situations.
For example, from the classification of line bundles over $\mathbb{P}_\mathbb{C}^1$, I can use the pullback of a connection and show that the degree/first Chern number is invariant under the action of $g\in SL(2,C)\subset Aut(\mathbb{P}_\mathbb{C}^1)$, and thus $g^*L\simeq L$ for every line bundle over the projective line.
I am trying to come up with a proof that either implements explicitly an identity-covering-vector-bundle-map or a proof involving Cech cohomology, but I would really prefer the former.
If this is false, I would like to build an explicit counterexample.
 A: 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 $\mathbb{Z}/2\mathbb{Z}$.
