Let $M$ be a (closed, smooth) manifold of dimension $d$. For $n$ a positive integer, fix $n$ points $x_1, \dots, x_n \in M$. The group of diffeomorphisms of $M$ that permutes the points $x_i$ surjects to the symmetric group $\Sigma_n$, at least for $d \geq 2$:

$$\Phi\colon \text{Diff}(M,\{x_1, \dots, x_n\}) \to \Sigma_n$$

Is it always true that $\Phi$ does not have a section for $n$ big enough?

How big do we have to choose $n$ for $M = S^d$? (The naive guess would be $d+3$.)