  We construct an analogue of the classical descent theory of
Colliot-Th\'el\`ene and Sansuc in which algebraic tori are replaced with finite
supersolvable groups. As an application, we show that rational points are dense
in the Brauer-Manin set for smooth compactifications of certain quotients of
homogeneous spaces by finite supersolvable groups. For suitably chosen
homogeneous spaces, this implies the existence of supersolvable Galois
extensions of number fields with prescribed norms, generalising work of
Frei-Loughran-Newton.
