Suppose $M$ and $N$ are $\mathbb{F}[G]$-modules over a field $\mathbb{F}$, and $G$ is a finite group.
Claim: Suppose there is a map $f: M \rightarrow N$ such that $f$ is a $\mathbb{F}$-vector space isomorphism, and $f$ is $G$-equivariant. Then $$H^*(G, M) \cong H^*(G, N)$$ as $\mathbb{F}$-vector space.
In other words, two group cohomology are isomorphic if they have isomorphic coefficients.
Is my statement above true? I feel like something is missing, but I have no clue. Thank you in advance.