Let $H\to G$ be a subgroup of a finite group, let $M$ be an $H$-module, $N$ be the induced $G$-module, by Shapiro lemma we have $H^i(G,N)\cong H^i(H,M)$. What is the map on the level of cochains? Given a cochain $G^i\to N$, what is the corresponding $H^i\to M$? (If we use inclusion $H^i\to G^i$, what is the map $N\to M$?)