Riemannian isometry subgroups dominate finite subgroups in self-diffeomorphism group

Statement

Let $M$ be a differential manifold and $G$ be a finite subgroup of the self-diffeomorphism group of $M$. Then, there exists a Riemannian metric on $M$ such that the self-isometry group of $M$ with respect to that metric contains $G$.

Proof

The idea is to average out. It is almost exactly the same proof which shows that the orthogonal group is finite-dominating in the general linear group.