Linear and smooth oriented equivalence of orthogonal representations of finite groups

Let $n\le 5$ be an integer, and let $\mathrm{\Gamma }$ be a finite group. We prove that if $\rho ,{\rho }^{\mathrm{\prime }}:\mathrm{\Gamma }\to O(n)$ 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 $G\subset O(4)$ is a finite group, then exactly one of the following is true: the elements of $G$ have a common invariant $1$-dimensional subspace in ${\mathbb{R}}^4$; some element of $G$ has no invariant $1$-dimensional subspace; or $G$ is conjugate to a specific group $K\subset O(4)$ of order $16$.