Suppose we have a compact manifold $M$ with the action of a compact group $G$. Consider the space of $C^l$ $G$-equivariant diffeomorphisms $\text{Diff}_G^{l}(M)$ with the $C^l$ topology and the space of $C^\infty$ $G$-equivariant diffeomorphisms $\text{Diff}_G^{\infty}(M)$ with the $C^\infty$ topology.

Then is the inclusion $\text{Diff}_G^{\infty}(M) \hookrightarrow \text{Diff}_G^{l}(M)$ a homotopy equivalence? If so what is a good reference where the proof is spelt out?

Suppose we have a compact manifold $M$ with the action of a compact group $G$. Consider the space of $C^l$ $G$-equivariant diffeomorphisms $\text{Diff}_G^{l}(M)$ with the $C^l$ topology and the space of $C^\infty$ $G$-equivariant diffeomorphisms $\text{Diff}_G^{\infty}(M)$ with the $C^\infty$ topology.

Question. is the inclusion $\text{Diff}_G^{\infty}(M) \hookrightarrow \text{Diff}_G^{l}(M)$ a homotopy equivalence? If so, what is a good reference where the proof is spelt out?