TITLE: What is this concept related to group actions?
QUESTION [8 upvotes]: Inspired by Group Equivariant Convolutional Networks by Taco Cohen and Max Welling I have been thinking about the following construction and I wonder where else it has been described and studied.
Suppose $G$ is a group and $X$ and $Y$ are sets with a $G$-action.  Let $f : X \to Y$, not necessarily $G$-equivariant.  Then we can cook up augmentations of $X$, $Y$ and $f$ ($X_G$, $Y_G$ and $f_G$ respectively) such that $f_G$ is $G$-equivariant.
The recipe is to define $X_G = G \times X$ with $G$-action $g(h, x) = (gh, gx)$ (resp. for $Y_G$) and $f_G : X_G \to Y_G$ by $f_G(g, x) = (g, gf(g^{-1}x))$.
Then we have that

$f_G(g(h, x)) = f_G(gh, gx)$
$= (gh, ghf((gh)^{-1}gx)$
$= (gh, ghf(h^{-1}g^{-1}gx))$
$= (gh, ghf(h^{-1}x))$
$= g(h, hf(h^{-1}x))$
$= gf_G(h, x)$

i.e. $f_G$ is $G$-equivariant.
It seems like this should be a well-known construction but I just can't quite grasp its implications at the moment. Is there somewhere I can read more about it?

REPLY [8 votes]: In fact $X_G=G\times X$ with the action $g(h,x)=(gh,gx)$ is $G$-equivariantly isomorphic to $G\times|X|=G\times X$ with the action $g(h,x)=(gh,x)$. The isomorphism $i_X:X_G\to G\times|X|$ is given by $i_X(h,x)=(h,h^{-1}x)$, with the inverse $i_X^{-1}(h,x')=(h,hx')$.
Now observe that for any set $S$, the $G$-set $G\times S$ with the action $g(h,s)=(gh,s)$ is the free $G$-set on $S$; thus any map $f:S\to Y$ to a $G$-set $Y$ extends uniquely to a $G$-equivariant map $\bar f:G\times S\to Y$ with $\bar f(1,s)=f(s)$, namely $\bar f(h,s)=hf(s)$.
It follows that for any map $f:X\to Y$ between $G$-sets, there is a unique $G$-equivariant map $\tilde f:X_G\to Y$ with $\tilde f(1,x)=f(x)$, namely, $\tilde f=\bar f\circ i_X$, i. e. $\tilde f(h,x)=hf(h^{-1}x)$:
$$
X_G\xrightarrow{i_X}G\times|X|\xrightarrow{\bar f}Y
$$
In the above notation your $f_G$ is $(\pi_G,\tilde f)$, where $\pi_G:X_G=G\times X\to G$ is the projection (which is obviously equivariant).