Does anyone know a reference, article, book, etc... that defines the transfer morphisms of group (co)homology on chain/cochain level?
I'm talking about the maps $$\text{tr}\colon H^*(H, -) \to H^*(G, -)$$ $$\text{tr}\colon H_*(G, -) \to H_*(H, -)$$ that arise when $[G\colon H] <\infty$ for a subgroup $H$ of a group $G$, and how to define them explicitly for the standard resolutions of $\mathbb{Z}$ over $G$ and over $H$:
$$\cdots \to \mathbb{Z}[G^{i+1}] \to \mathbb{Z}[G^i] \to \mathbb{Z}[G^{i-1}]\to \cdots \to \mathbb{Z}[G] \to \mathbb{Z} \to \{0\}$$ $$\cdots \to \mathbb{Z}[H^{i+1}] \to \mathbb{Z}[H^i] \to \mathbb{Z}[H^{i-1}]\to \cdots \to \mathbb{Z}[H] \to \mathbb{Z} \to \{0\}$$
You can have a look at [1, Chapter I.§§5, pag 48] or in [2, Proposition 2.5.2].
The book [2] contains a full section (Section 2.5) devoted to the derivation of explicit formulas for cohomology maps. The author obtains a formula arguing by induction. He does in detail the first three steps, enough to leave the rest as an exercise to the reader.
The book [1] contains much more material. The authors define two maps: one is the corestriction defined abstractly, the other is the map given by the formula. They argue that both are homomorphisms of complexes, and they are functiorial. Then that they commute with the $\delta$-homomorphism. Since they also agree on dimension 0, they are equal by induction (dimension shifting).
[1] Neukirch, Jürgen; Schmidt, Alexander(D-RGBGNS1); Wingberg, Kay(D-HDBG) Cohomology of number fields. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 323. Springer-Verlag, Berlin, 2008. xvi+825 pp. ISBN: 978-3-540-37888-4
[2] Weiss, Edwin Cohomology of groups. Pure and Applied Mathematics, Vol. 34 Academic Press, New York-London 1969 x+274 pp.