Let $R$ be a commutative ring, $f:H\to G$ a surjective group homomorphism and consider $RG$ as a $(RG,RH)$-module via $g\cdot h := g\cdot f(h)$ as usual. Now suppose that $RG$ is flat over $H$, meaning that $$RG\otimes -:H\text{-}\mathbf{Mod} \to G\text{-}\mathbf{Mod}$$ is exact. What can we say about $f$?
If $RG$ was even projective over $H$, we would get a section $s:RG\to RH$ telling us that $\#\mathrm{ker}(f)<\infty$ is a unit in $R$, but I suppose, something like this does not work if $RG$ is only assumed to be flat?