I would like to know examples (with references, if possible) of the following:
(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ but admitting no effective linear action on $S^n$ (so $G$ is not isomorphic to any subgroup of $O(n+1,\mathbb{R})$);
(2) a finite group $G$ acting effectively and smoothly on $\mathbb{R}^n$ (any $n$) but admitting no effective action on $\mathbb{R}^n$ by affine transformations;
(3) a finite group $G$ acting effectively and smoothly on the closed $n$-dimensional disk $D^n$ (any $n$), but admitting no effective linear action on $D^n$ (so $G$ is not isomorphic to any subgroup of $O(n,{\mathbb R})$).
Some comments on these questions:
A paper of Petrie ("Free metacyclic group actions on homotopy spheres") gives an example of finite group acting freely on a sphere $S^n$ but admitting no free linear action on $S^n$.
For $p$-groups, there are no examples of (1) and (3), by a theorem of Dotzel and Hamrick.
An example of (3) is automatically an example of (2), since a finite group of affine transformations of ${\mathbb R}^n$ fixes a point and hence can be identified with a finite subgroup of $O(n)$.
There are in the litterature several results implying that there are no low dimensional examples of (1), (2) and (3).