Linear and smooth oriented equivalence of orthogonal representations of finite groups

Let n≤ 5 be an integer, and let Γ be a finite group. We prove that if ρ , ρ': Γ → O(n) are two representations that are conjugate by an orientation-preserving diffeomorphism, then they are conjugate by an element of $SO(n)$. In the process, we prove that if G ⊂ O(4) is a finite group, then exactly one of the following is true: the elements of G have a common invariant $1$-dimensional subspace in R⁴; some element of G has no invariant $1$-dimensional subspace; or G is conjugate to a specific group K ⊂ O(4) of order $16$.