Linear and smooth oriented equivalence of orthogonal representations of finite groups

Let n 5 be an integer, and let be a finite group. We prove that if, ': 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.