Let $ SU_2 $ be the group of 2 by 2 unitary matrices with determinant 1. $ SU_2 $ contains the binary icosahedral group $ I^* $ as a closed subgroup. Since $ SU_2 $ is compact and $ I^* $ is a closed subgroup then by Mostow-Palais theorem there exists a representation $ \pi: SU_2 \to GL(V) $ and some $ v \in V $ such that stabilizer of $ v $ $$ \{g \in SU_2: \pi(g)v=v \} $$ is exactly $ I^* $. And thus the orbit of $ v $ $$ \mathcal{O}_v:=\{ \pi(g)v : g\in SU_2 \} $$ is exactly $ SU_2/I^* $, which is the Poincare homology sphere.
The same story is true with $ SO_3(\mathbb{R}) $ and the icosahedral group $ I $. Since $ SO_3(\mathbb{R}) $ is compact and $ I $ is a closed subgroup then again by Mostow Palais there exists some representation $ \pi: SO_3(\mathbb{R}) \to GL(V) $ and some $ v \in V $ such that the stabilizer of $ v $ is $ I $ and the thus the orbit of $ v $ is $ SO_3(\mathbb{R})/I $, which is again the Poincare homology sphere.
I am looking for some representation $ \pi $ and vector $ v $ such that the orbit of $ v $ is the Poincare homology sphere.
What I have tried so far:
(1) The identity irrep of $ SU_2 $ on $ \mathbb{C}^2 $. All the nonzero vectors have trivial stabilizer and have orbit $ S^3$.
(2) The identity irrep of $ SO_3(\mathbb{R}) $ on $ \mathbb{R}^3 $. All nonzero vectors have stabilizer a circle and orbit a 2 sphere.
(3) There is an irrep of $ SO_3(\mathbb{R}) $ on the 5 dimensional real vector space of traceless symmetric real 3 by 3 matrices given by conjugation $$ \pi(g)M=gMg^{-1} $$ but since every symmetric matrix is orthogonally diagonalizable every orbit is the orbit of a diagonal matrix and thus either $ \mathbb{RP}^2 $ when there is a repeated eigenvalue or it is the prism manifold with fundamental group the 8 element quaternion group when every eigenvalue is distinct.
Final note:
Although I'm most interested in actually finding the representation $ \pi $ and the vector $ v $, I would also be interested in bounds on the dimension of the such a $ \pi $. Like some sort of equivariant Whitney embedding theorem that says for a manifold $ M $ of dimensions $ n $ with a transitive compact group action then there always exists a representation of dimension $ 2n $ or less which realizes $ M $ as the orbit of some vector.
The representation of $ SO_3 $ is on the 13 dimensional real vector space of homogeneous degree 6 polynomials in three real variables $ x,y,z $. The vector with stabilizer the icosahedral group (and thus orbit the Poincare homology sphere) is the harmonic polynomial $$ 21(2-\sqrt{5})(x^2-\phi^2y^2)(y^2-\phi^2z^2)(z^2-\phi^2x^2)+(x^2+y^2+z^2)^3 $$ where $ \phi $ is the golden ratio. 13 is the minimal dimension for which such an orbit arises.