Suppose $G$ is a finite group and that $\rho: G\rightarrow O(d)$ is an orthogonal representation, with action on $\mathbb{R}^d$ denoted $\cdot$, such that $\rho_g$ is the identity only if $g=e$. Let's say that $\rho$ is "strongly" angle preserving if it is not trivial and

\begin{equation}\langle g\cdot v, v\rangle = \langle g\cdot w, w\rangle \end{equation} for all $v, w$ on the (representation) sphere $S^{d-1}$. (There must be a name for such an action, but let's stick with this name and let someone edit it as desired).

Examples of such representations would be all the finite subgroups of the classical algebras $\mathbb{R},\mathbb{C},$ and $\mathbb{H}$ under their standard embeddings in $O(1)$, $U(1)$, and $Sp(1)$.

Question: Can anyone point me towards literature on any other groups admitting such a representation?

Of course, these representations give free linear acts on spheres ($g\cdot v=v$ forces $\langle gw, w\rangle =1$ for all $w\in S^{d-1}$), and these are all known.