Strong approximation for homogeneous spaces of linear algebraic groups

Building on work by Yang Cao, we show that any homogeneous space of the form $G/H$ with $G$ a connected linear algebraic group over a number field $k$ satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some natural compactness assumptions when $k$ is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form $G/H$ with $G$ semisimple simply connected and $H$ finite, using the theory of torsors and descent. (This latter result is somewhat related to the Inverse Galois Problem.)