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It's enough to pick a contractible manifold $M$ with two non-homotopic actions. For example, let us pick $M=\mathbb{R}^n$ with two actions of $C_2$, the trivial one and one given by a reflection. These two actions are not homotopic, since in the nontrivial action, the nontrivial element of $C_2$ is sent to a nonzero component of $\mathrm{Diff}(M)$ (if you want, it is not orientation-preserving), but of course all continuous maps with target $M$ are homotopic.