Let $G$ be a group, and let $H$ be a subgroup of index $m$. Let $A$ be a $G$-module. we have restriction $$\mathrm{Res}: H^n(G,A)\to H^n(H,A)$$ and co-restriction $$\mathrm{Cor}: H^n(H,A)\to H^n(G,A).$$ It is known that $$\mathrm{Cor}\circ \mathrm{Res}(c)=mc.$$ I don't see why the converse holds:
Why $$\mathrm{Res}\circ \mathrm{Cor}(c)=mc$$ for every $c\in H^n(H,A)$?
In this direction applying Res doesn't do anything to the function, and we stay with some sum $\sum_{i=1}^m g_i f(g_i^{-1}p)$ for every $p\in P_n$ for a projective resolution $P_n$ of $\mathbb Z$ as a $G$-module. The different terms in the sum seem like distrinct functions, and we cannot put the $g_i^{-1}$ outside because $f$ is only a $\mathbb Z H$-module homomorphism.
Here's the claim in Dummit & Foote:
You say that $f$ is only an $H$-module homomorphism, but in fact it's a $G$-equivariant: we started with a $G$-invariant homomorphism, "forgot" it was $G$-invariant, and then applied the formula with the sums. Since it was $G$-invariant, each term in the formula with the sum is just $f(p)$.
EDIT: as discussed in the comments, the other claim about corestricting and then restricting is false. Reference Neukirch Schmidt and Wingberg Corollary 1.5.7 for the correct formulation.