I am trying to understand Proposition 4.1 of Nakajima's paper 'Invariants of finite groups generated by pseudo-reflections in positive characteristic' (link here: https://pdfs.semanticscholar.org/da5f/8e1b4adc0fbb0141491c6aa6330123d0fc47.pdf).
Statement: Let $k$ be a field, $G$ a finite group (here $k$ has arbitrary characteristic $p > 0$: in particular I'm interested in the case $|G| \equiv 0 $ mod $p$). Let $V$ and $W$ be finite-dimensional $G$-faithful $kG$-modules, and let $\phi: V \to W$ be a $kG$-linear epimorphism. If $k[V]^G$ is polynomial, then $k[W]^G$ is polynomial.
Proof: Write $k[V] = \sum_{i=1}^{|G|} k[V]^G f_i$ for some $f_i \in k[V]$. Let $\widetilde{\phi}: k[V] \to k[W]$ be the induced epimorphism. Then $k[W] = \sum_{i=1}^{|G|} k[W]^G \widetilde{\phi}(f_i)$. Nakajima then claims that since $G$ acts faithfully on $W$, $k[W]$ is a free $k[W]^G$-module. It is this claim that I don't currently understand. This statement then implies that $k[W]^G$ is polynomial.
Any help or references would be appreciated.