  We study the relation between the structure of algebraic and context-free
subsets of a group G and that of a finite index subgroup H. Using these
results, we prove that a kind of Fatou property, previously studied by Berstel
and Sakarovitch in the context of rational subsets and by Herbst in the context
of algebraic subsets, holds for context-free subsets if and only if the group
is virtually free. We also exhibit a counterexample to a question of Herbst
concerning this property for algebraic subsets.
