  We relate the existence of some surfaces of general type and maximal Albanese
dimension to the existence of some monodromy representations of the braid group
$\mathsf{B}_2(C_2)$ in the symmetric group $\mathsf{S}_n$. Furthermore, we
compute the number of such representations up to $n=9$, and we analyze the
cases $n \in \{2, \, 3, \, 4\}$. For $n=2, \, 3$ we recover some surfaces with
$p_g=q=2$ recently studied (with different methods) by the author and his
collaborators, whereas for $n=4$ we obtain some conjecturally new examples.
