Smooth Maps to Set of Conjugacy classes

Question

Let $G$ be a Lie group and $M$ be a manifold. Let $\mathrm{C}\left(G\right)$ be the set of conjugacy classes of $G$ and $C^\infty\left(S^1,M\right)$ be the class of smooth maps $S^1\to M$. Are there smooth structures on either $\mathrm{C}\left(G\right)$ and $C^\infty\left(S^1,M\right)$? If yes, is there a good characterisation of smoothness for a map $W\colon C^\infty\left(S^1,M\right)\to\mathrm{C}\left(G\right)$? In particular, can I find the second derivation of $W$ with respect to the constant map to a point $x$?