My previous question (where $n=2$) was a bit too naive. I think that this one, which is the one being of genuine interest to me, is more involved.

Let $f=\mathbb R^n\rightarrow\mathbb R$ be a symmetric function: $$f(x^\sigma)=f(x),\qquad\forall\sigma\in\frak S_n,$$ where $x^\sigma:=(x_{\sigma(1)},\ldots,x_{\sigma(n)})$. There exists a unique function $F$, defined on the image of $\mathbb R^n$ under the map $x\mapsto (p_1,\ldots,p_n)$, where $p_j$ is the elementary symmetric polynomial of degree $j$, such that $$f(x)=F(p_1,\ldots,p_n).$$

What is the regularity of $F$ ? Does $f\in\mathcal C^r$ imply $F\in\mathcal C^{r/n}$ and not more ?

*Edit*. By $\mathcal C^s$-regularity on a closed domain, one means as usual that the function under consideration is the restriction of a $\mathcal C^s$-function defined on an open super-set.