Let $G$ be a finite group, let $H$ be a subgroup of $G$, let $W$ be an $H$-module and let $V$ be the $G$-module induced from $W$. Shapiro's lemma says that $H^1(G,V)\cong H^1(H,W)$. I was wondering if there exists an explicit description of this isomorphism at least in the case that $W$ is the trivial module and hence $V$ is a permutation module.
Ideally, I would like a description where given an element of $H^1(H,W)$, I can explicilty describe the corresponding element in $H^1(G,V)$ using a transversal of $H$ in $G$.
