  Let $n\le 5$ be an integer, and let $\Gamma$ be a finite group. We prove that
if $\rho , \rho': \Gamma \to O(n)$ 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 $16$.
