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 $s\colon H\backslash G\to G$ of the projection $G\to H\backslash G$ onto the quotient space $H\backslash G$.
Consider the homomorphism 
$$ N\colon M\to M,\quad\ m\mapsto \sum_{x\in H\backslash G} s(x) m.$$
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
$$ \sum_{x\in H\backslash G} s(x)\cdot( g m -m)$$
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)?