What are concrete examples of fibre bundles with fibre genus $\geq 2$? I am trying to find examples I can use to work throught the following construction: 


Suppose $X \to C$ is a fibre bundle with $C$ a curve and for which the fibre $F$ is a curve of genus $\geq 2$. Then $\mathrm{Aut}(F)$ is a finite group and we can look at $\bar{F} := F/\mathrm{Aut}(F)$. We get a natural morphism $X \to \bar{F} \times C$. We can compose this with Belyi morphisms for $\bar{F}$ and $C$ to obtain a cover of $\mathbb{P}^1 \times \mathbb{P}^1$. 


Hence with 'concrete' I mean that by following the above construction I can actually see what the resulting cover of $\mathbb{P}^1 \times \mathbb{P}^1$ looks like.



Fibre bundles of fibre genus greater than 1
All such fiber bundles are constructed in the following way : choose an étale covering  

$\tilde{C}\rightarrow C $ with Galois group $G\subset \mathrm{Aut}(F)$, and take $S=(\tilde{C}\times F )/G$, with $G$ acting on each factor in the obvious way. The fiber bundle structure is given by the first projection $S\rightarrow C$. This is quite concrete.


