A transitive set in $\mathbf{R}^n$ is a finite set with a transitive group of symmetries. I want to understand how subsets of a transitive set constrain the group.

Let me start with the example of a tetrahedron. A tetrahedron has symmetry group isomorphic to $S_4$, which is of course nonabelian. However, it is possible to embed the tetrahedron into a larger set, the cube, which has a transitive abelian group of symmetries isomorphic to $C_2^3$. (Of course, the full symmetry group of the cube is $C_2 \times S_4$, which is nonabelian, but my point is there is an abelian subgroup which is still transitive.) So we made things simpler by going bigger.

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[pictures borrowed from Wikipedia]

My specific question is the same with (tetrahedron, abelian) replaced with (icosahedron, soluble).

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Is there a finite set in $\mathbf{R}^n$ (with possibly $n > 3$) with a transitive soluble group of symmetries containing the vertices of a regular icosahedron?

I would also appreciate any references to related matters.