If $H$ is a finite index normal subgroup of $G$, then $H^*(G; \mathbb{Q}) = H^*(H; \mathbb{Q})^{G/H}$. This follows from some well-known spectral sequence (correct?)
What does the notation of exponentiation by $G/H$ mean?
Is it possible for $H^*(G; \mathbb{Q}) = 0$ but $H^*(H; \mathbb{Q}) \neq 0$? An easy example (for $H^1$ or $H^2$)?