This is a continuation of my previous question. Let $R$ be a commutative ring in which $n!$ is invertible. Does it follow that every $R \Sigma_n$-module decomposes as a direct sum of modules of the form $V_i \otimes_R M_i$, where $M_i$ is some $R$-module and $V_i$ is some $R \Sigma_n$-module which is a direct summand of $R \Sigma_n$?
Notice that, even if we had $R \Sigma_n \cong \prod_i M_{n_i}(R)$ (is this true?), and $V_i$ is chosen to be $W_i \otimes_{\mathbb{Z}} R$, where $W_i$ is some integral model for a irreducible complex representation $W_i \otimes_{\mathbb{Z}} \mathbb{C}$ of $\Sigma_n$, the usual projection $\mathbb{C} \Sigma_n \to W_i \otimes_{\mathbb{Z}} \mathbb{C}$ does not seem to be definable over $\mathbb{Z}[1/n!]$, and therefore it is not clear to me if $V_i$ is a direct summand of $R \Sigma_n$.