  We provide a description of the fundamental group of the quotient of a
product of topological spaces $X_i$, each admitting a universal cover, by a
finite group $G$, provided that there is only a finite number of path-connected
components in $X_i^g$ for every $g\in G$. This generalizes previous work of
Bauer-Catanese-Grunewald-Pignatelli and Dedieu-Perroni.
