Real approximation for homogeneous spaces with finite stabilizers

We prove some new cases of real appoximation for homogeneous spaces with finite stabilizers and describe the state of the art around this question, giving proofs that are well-known to experts but that, to our knowledge, cannot be found in the literature. Our main new result needs the latest advances in the topic of the Brauer--Manin obstruction for homogeneous spaces with supersolvable stabilizers. It states that any finite $k$-group that is split by a $2$-primary extension satisfies real approximation.