Linear and smooth oriented equivalence of orthogonal representations of finite groups

Let Γ be a finite group. We prove that if $\rho ,{\rho }^{\prime }:\Gamma \to O\left(4\right)$ are two representations that are conjugate by an orientation-preserving diffeomorphism of $S^3$, then they are conjugate by an element of $\mathrm{SO}\left(4\right)$. In the process, we prove that if $G\subset O\left(4\right)$ is a finite group, then exactly one of the following is true: the elements of G have a common invariant 1-dimensional subspace in $R^4$; some element of G has no invariant 1-dimensional subspace; or G is conjugate to a specific group $K\subset O\left(4\right)$ of order 16.