The Grunwald problem and homogeneous spaces with non-solvable stabilisers

We give an affirmative answer to the Grunwald problem for new families of non-solvable finite groups G, away from the set of primes dividing |G|. Furthermore, we show that such G verify the condition (BM), that is, the Brauer-Manin obstruction to weak approximation is the only one for quotients of SLₙ by G. These new families include extensions of groups satisfying (BM) by kernels which are products of symmetric groups Sₘ, with m 2, 6, and alternating groups A₅. We also investigate (BM) for small groups by giving an explicit list of small order groups for which (BM) is unknown and we show that for many of them (BM) holds under Schinzel's hypothesis.