In the proof of Maschke's theorem, we magically transform linear transformations $T$ into maps that preserve the group ring structure. The method is to define
$ \bar T(v) = \frac{1}{|G|} \sum\limits_{g \in G} g^{-1} T(g(v))$
Note for this to make sense, we have to be able to divide by $ |G|$. In other words, the characteristic of $ F$ does not divide $ |G|$. When $ T$ is already an $ FG$-module homomorphism:
$ \bar T(v) = \frac{1}{|G|} \sum\limits_{g \in G} g^{-1} T(g(v)) = \frac{1}{|G|} \sum\limits_{g \in G} g^{-1} gT(v) = \frac{1}{|G|} \sum\limits_{g \in G} T(v) = T(v)$
In essence, we would like $ g^{-1}T(g(v))$ to just be $T(v)$ for any linear transformation $ T$, but it's not. So we form a function by taking averages and trying to water down the deviation. And it works! We end up with an honest $ FG$-module morphism.
I saw another example of this in Guillemin/Pollack's Differential topology. Here they start with a n-tensor $T$ and form an alternating tensor:
$Alt(T) = \frac{1}{n!} \sum\limits_{g \in S_n} sgn(g) T^g$
My question is: Where does this kind of "averaging to pick up structure" work? Can we turn set maps into equivariant maps in the G-set setting? Or something different like averaging continuous maps to get smooth ones. The obvious obstruction is being able to divide, so is there a way around this? So a second question is: when you can't do this averaging, what kind of workarounds exist?