Transfer for the group of coinvariants: a reference request

Question

Let $G$ be a group and $M$ be a $G$-module, that is, an abelian group written additively on which $G$ acts: $$ (g,m)\mapsto g m.$$ We consider the group of coinvariants $$ M_G:=G/\langle g m -m\ |\ g\in G,\,m\in M\rangle. $$ Let $H\subseteq G$ be a subgroup of finite index. I am trying to understand the transfer map $ M_G\to M_H$.

Choose a section $s\colon H\setminus G\to G$ of the projection $G\to H\setminus G$ onto the quotient space $H\setminus G$. Consider the homomorphism $$ N\colon M\to M,\quad\ m\mapsto \sum_{x\in H\setminus G} s(x) m.$$ This homomorphism $N$ induces a homomorphism $$ N_*\colon M\to M_H, $$ which is clearly independent of the choice of the section $s$.

Question 1. Why does the homomorphism $N_*$ descend to a homomorphism $$ N_{**}\colon M_G\to M_H\ ?$$

In other words, for $m\in M$ and $g\in G$, why is $$ \sum_{x\in H\setminus G} s(x)\cdot( g m -m)$$ a linear combination of elements of the form $h m'-m'$ for $h\in H$ and $m'\in M$?

I have asked Question 1 in Mathematics Stack Exchange https://math.stackexchange.com/q/4833018/37763 and got an excellent answer by darij grinberg. Now, in my paper I can refer to this answer. However, if possible, I would like to refer to a book or a paper.

Question 2. What are possible references to an answer to Question 1 (in addition to the answer in MSE)?

Answer 1

Let's take some care about left vs. right modules, and then use associativity and functoriality of the tensor products for the associative $\mathbb{Z}$-algebras $\mathbb{Z}G$ and $\mathbb{Z}H$.

Write $\underline{\mathbb{Z}}$ for the integers with a trivial right group action. The coinvariants of the left module $M$ can be thought of as $$ M_G = \underline{\mathbb{Z}} \otimes_G M. $$ Restriction to the action of $H$ can similarly be written as a tensor product with the bimodule ${}_H G_G$, so we also get a formula for $M_H$: $$ M_H = \underline{\mathbb{Z}} \otimes_H G \otimes_G M. $$ The transfer map $M_G \to M_H$ is functorial in $M$, suggesting that the map acts only on the tensor factor preceding the $M$. And indeed, we induce the transfer using a map of right $G$-modules

$$ \underline{\mathbb{Z}} \to \underline{\mathbb{Z}} \otimes_H G = G_H. $$ The coinvariants of $H$ acting on the left of $G$ are cosets. As there is a single orbit, any $G$-equivariant map from $\underline{\mathbb{Z}}$ is a multiple of the map sending $1$ to the sum of the cosets.