In what follows all the groups will be discrete, not necessarly finite.

Let $f:G\to H$ be a morphism of groups and $H'\to H$ be the inclusion of a subgroup. It seems to me (but correct me if I am wrong) that the pullback $BG\times_{BH}BH'$ is $B(f^{-1}H)$.

If this is true, is there a more general statement that tells you when a commutative square of classifying space is a pullback diagram?