Suppose that $G$ is a group acting on a commutative ring $R$, inducing an action on each $R$-module. For any $R$-module $M$, let $M^G$ denote the collection of elements of $M$ invariant under the $G$-action. Then, $M^G$ is an $R^G$-module.

Here is my question. Do we have:

$$ (M\otimes_RN)^G=M^G\otimes_{R^G}N^G $$

for $R$-modules $M$ and $N$?

At least does it hold if $M$ and $N$ are projective? Clearly, this holds when they are free modules.

Are there conditions on the group $G$ that could create this situation? For example, $G$ could be a finite group or an abelian group or a linear reductive group... etc etc

Or perhaps, there could be conditions on the ring $R$ for which we get the above equality?