Let $\Delta$ be a simplicial sphere, that is, a finite (abstract) simplicial complex whose canonical geometric realization $|\Delta|$ is homeomorphic to a sphere $\mathbf S^d\subset\Bbb R^{d+1}$.

Question:Can the homeomorphism $\phi :|\Delta|\to\mathbf S^d$ be chosen in a way, so that all combinatorial symmetries of $\Delta$ are realized geometrically?That is, if $\sigma :\Delta\to\Delta$ is a combinatorial automorphism of $\Delta$ (a bijective simplicial map) I want there to be an isometry $\smash{f_\sigma:\mathbf S^d\to\mathbf S^d}$ so that $$\phi\circ \sigma = f_\sigma\circ \phi.$$

You can think of this as a subdivision of $\mathbf S^d$ that has the same symmetries as the abstract simplicial complex $\Delta$. If we consider the sphere embedded in $\smash{\Bbb R^{d+1}}$, the isometries are exactly the orthogonal transformations restricted to $\smash{\mathbf S^d}$.