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

I was inspired to ask by this question: Example of a manifold which is not a homogeneous space of any Lie group

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

I was inspired to ask by this question: Example of a manifold which is not a homogeneous space of any Lie group

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: 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. Could someone explain that?

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

I was inspired to ask by this question: Example of a manifold which is not a homogeneous space of any Lie group

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: 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?

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: 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. Could someone explain that?