This is an example of Rotman Homological Algebra 5.72 (2).
$S^1\subset C$ is unit circle where $C$ is complex plane. Let $p:S^1\to S^1$ be $p:z\to z^2$ and let $\operatorname{dom}(p)=E, \operatorname{codom}(p)=X$. It is clear that $X$ has a double cover sitting above through $E\to X$ projection by $p$. It is clear that stalks of $E$ are isomorphic to $Z_2$ abelian groups. Furthermore $E$ is Etale sheaf over $X$. It is clear that sheaf of sections has same stalks as $E$.
How do I see sheaf of sections $\Gamma(-,E)$ of $E$ not isomorphic to $E$?
The book is saying $\Gamma(-,E)$ is not isomorphic to constant $Z_2$ sheaf over $X$. This is clear.
Hint : $ \Gamma(E,E)$ contains the identity element and also the map which switch the two points in the fiber, on the other hand $E$ considered as Etale-sheaf over $X$ has no global sections as it would gives a section of the map $z \mapsto z^2$.