Abstract: The inverse Galois problem has many refinements. In this talk, we focus on the existence of abelian Galois extensions with restricted ramification. The Gras--Munnier formula provides a quick test for whether such extensions exist. We discuss how the Gras--Munnier formula corresponds to a Brauer--Manin obstruction to strong approximation on the classifying stack BG. This is joint work with Daniel Loughran.