Finitely generated groups of diffeomorphisms with prescribed critical regularity in every dimension

Let $n\ge 2$, and let $M$ be either the $n$-dimensional sphere ${\mathbb{S}}^n$ or the real projective space ${\mathbb{R}\mathbb{P}}^n$. Given $\alpha \ge 1$, we construct a finitely generated group $G_{n,\alpha }$ that embeds into the group of $C^{\alpha }$-diffeomorphisms of $M$, but does not embed into the group of $C^{\beta }$-diffeomorphisms of $M$ for any $\beta >\alpha$.