We construct an analogue of the classical descent theory of Colliot-Thélène 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.