When a homogeneous space $G/H$ is given, a differential geometer may ask
Is this homogeneous space also Riemannian homogenous?
Now many sources I read interpret this question as whether this space admits a $G$-invariant metric on $G/H$, but I imagine it is possible that while we could not find a $G$-invariant metric on $G/H$, there still may be some mysterious $G'$ that $$G'/H' \cong G/H$$ where we may find $G'$-invariant metric on $G'/H'$. Here is my doubt,
- Why is this non-uniqueness of representing $G/H$ rarely mentioned in text?
To be fair, a few sources I read have brought up this issue, but they all seem to assume some inclusion $G' \hookrightarrow G$ and say we may use third isomorphism to tidy the quotient up. However, it still appears strange to me, since it is possible that there is no any relation between $G$ and $G'$, or is it?