Given any manifold $ M $ does there exist  $ G $ a Lie group and $ H,\Gamma $ closed subgroups of $ G $ such that 
$$
M \cong (G/H)/\Gamma
$$

I was inspired to ask by this question:  https://mathoverflow.net/questions/89345/example-of-a-manifold-which-is-not-a-homogeneous-space-of-any-lie-group

The answer by Vitali Kapovitch claims that 
$$
(S^3\times S^3) \# (S^3\times S^3)
$$
cannot be a biquotient space. My question is a bit more general than biquotient space, but perhaps this is still a counter example?