Are there simplicial spheres with "non-geometric symmetries"? 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 $\smash{\mathbf S^d}$.
 A: The answer is negative. Already in dimension 4 there are fake real-projective spaces, which are smooth 4-manifolds homotopy-equivalent but not homeomorphic to $RP^4$. These correspond to smooth free involutions $\sigma: S^4\to S^4$ which are not topologically conjugate to orthogonal transformations. Similar examples exist in higher dimensions. See manifold atlas for references.
However, in dimensions $n\le 3$ indeed, every finite group of PL homeomorphisms of $S^n$, is PL conjugate
to a finite subgroup of the orthogonal group. This is easy in dimension 2 and hard in dimension 3 (a consequence of the "orbifold geometrization theorem").
Edit. A nice, although dated, survey is
M.Davis, A survey of results in higher dimensions, In "The Smith Conjecture", (editors: J. W. Morgan and H. Bass), Academic Press, New York, 1984.
dealing with examples of "exotic" actions of compact (in particular, finite) groups on spheres. Some of the examples he discusses are PL.  Note that smooth actions of finite groups can be always made PL.
