What is the name of this function on group-ring modules? Let $G$ be a finite group. Let $M$ be a $\Bbb{Z}G$-module. 
What is the name of the map $M\to M$ given by:
$$
m \mapsto \sum_{g\in G} g m\, ,
$$
possibly divided by $|G|$?
What is it used for?
A reference would be also welcome.
Thanks!

Possibly divided by $|G|$, I would call it "averaging over $G$." I'm not sure it has a single widely-agreed-upon name. Sometimes people call it the Reynolds operator. 
The key property of averaging is that it lands in $G$-invariants: that is, any element of the form $\sum_{g \in g} gm$ is invariant under the action of $G$. If you are allowed to divide by $|G|$, then it's furthermore the case that $G$-invariants are fixed by this map. So you conclude that the submodule $M^G$ of $G$-invariants is a direct summand. This is a basic fact and has several important consequences, such as 1) Maschke's theorem, and 2) the vanishing of group cohomology over $\mathbb{Z} \left[ \frac{1}{|G|} \right]$. 
