$\sigma(gv_1\otimes\ldots\otimes gv_n)=gv_{\sigma^{-1}(1)}\otimes\ldots\otimes gv_{\sigma^{-1}(n)}$?

Question

Let $V$ be a $\mathbb{C}G$-module, then $V^{\otimes n}$ can be endowed with a $\mathbb{C}G$-module structure via $g(v_1\otimes\ldots\otimes v_n)=gv_1\otimes \ldots \otimes gv_n$. Now consider $Sym^n(V)=\{w\in V^{\otimes n}| \sigma(w)=w \forall \sigma \in S_n\}$, where $\sigma(w_1\otimes \ldots \otimes w_n)= w_{\sigma^{-1}(1)}\otimes\ldots \otimes w_{\sigma^{-1}(n)}$. I have to show that $Sym^n(V)$ is a submodule. I can almost show it but I am missing the argument that says that $\sigma(gv_1\otimes\ldots\otimes gv_n)=gv_{\sigma^{-1}(1)}\otimes\ldots\otimes gv_{\sigma^{-1}(n)}$.