For a seminar on group cohomology I want to avoid having the students talk about group rings and resolutions for the time being and do everything with explicit inhomogenous cochains.
Now let $A$ be a $G$ module where $G$ is a (finite) group. Let $H$ be a subgroup (not necessarily normal). Let $CoInd_H^G(A)=Map(G/H,A)$ be the coinduced module with the $G$ action given by $(g.f)(x)=g.(f(g^{-1}x))$ for $x\in G/H$.
Evaluation at $H$ gives a map $p:Map(G/H,A)\rightarrow A$.
Given a cochain $f:G^n\to CoInd_H^G(A)$ we obtain a map $p\circ f|_H:H^n\to A$. This induces $H^n(G,CoInd_H^G(A))\rightarrow H^n(H,A)$. I want to see that this is an isomorphism, but I cannot find the inverse.
D. Naidu constructs an explicit inverse map $H^n(H,A) \rightarrow H^n(G,CoInd_G^H(A))$ for the case $n=1,2$. For the construction and proof, check out lemma $2.1, 2.2$ in his paper https://arxiv.org/pdf/math/0605530.pdf.