Let $G$ be a compact second countable Hausdorff group, and let $X=G/H$ be a homogeneous space with $H\subset G$ a closed subgroup. Let further $K\subset G$ be another closed subgroup.
Does there exist a section $s:K\backslash X\to X$ from the orbit space (double coset space) $K\backslash X=K\backslash G/H$ to the homogeneous space $X$ ( i.e., $Ks(Kx)=Kx$ for all $x\in X$ ) which is continuous aside from at most countably many orbits $Kx$?
How does the answer to Questoin 1 change if I assume that $G$ is Lie?
Even if $G$ is Lie, $X\to K\backslash X$ is not really a geometrical fibre bundle, since not all fibres ($K$-orbits) are isomorphic.