The automorphism group of a compact group action

This paper contains results on the structure of the group, $\operatorname {Diff} _G^r(M)$, of equivariant ${C^r}$-diffeomorphisms of a free action of the compact Lie group $G$ on $M$. $\operatorname {Diff} _G^r(M)$ is shown to be a locally trivial principal bundle over a submanifold of ${\operatorname {Diff} ^r}(X),X$ the orbit manifold. The structural group of this bundle is ${E^r}(G,M)$, the set of equivariant ${C^r}$-diffeomorphisms which induce the identity on $X$. ${E^r}(G,M)$ is shown to be a submanifold of ${\operatorname {Diff} ^r}(M)$ and in fact a Banach Lie group $(r < \infty )$.