Quotient of cartesian product by the right action of a group I've been recently reading about Burnside rings and I found Serge Bouc's paper. In one of its sections he explains different kinds of functors that will be considered in further reasoning. I got stuck at induction functor. Let me quote a fragment so that you see what I have in mind. 

Now if $Z$ is an $H$-set, the induced  $G$-set ${\rm Ind}_{H}^{G}Z $ is defined as $G \times_{H} X$, i. e. the quotient of the cartesian product $G \times X$ by the right action of $H$ given by $(g, x).h = (gh, h^{-1}x)$ for $g \in G, h \in H, x \in X$.

Okay, now a few words of explanation: $X$ is a $G$-set, $G$ is a group and $H$ is its subgroup.
Now, the problem is: what exactly is the quotient of cartesian product by the right action? How should I look at it? Any idea?
 A: In general if you start with a transitive $G$-set $\overline{X}$, an $H$-orbit $X$ may be a proper subset. Indeed, if we pick an element $x\in X$, in general $gx\not\in X$ if $g\in G\setminus H$. Conversely if we start with an $H$-set $X$, constructing the most "natural" $G$-set from it may require creating a larger set including new elements "$gx$" that don't already simplify according (we know how to simplify an expression $hx$ according to $H$'s action if $h\in H$).
To start with, let's represent the new element "$gx$" as a pair $(g,x)$. This is ill-defined however, since multiple pairs could represent the same thing. Specifically, $(h,x)$ ought to be $(e,hx)$, and more generally we ought to say that $(gh,x)=(g,hx)$ for all $g\in G,h\in H,x\in X$. This defines an equivalence relation on $G\times X$, and once we quotient it we get a $G$-set containing $X$ as an $H$-orbit.
The equivalence relation $(gh,x)\sim(g,hx)$ is the same as saying $(gh,h^{-1}x)\sim(g,x)$, and $(gh,h^{-1}x)$ defines a right action of $H$ on $G\times X$, so we see $G\times_HX$ is actually the quotient space (set of right $H$-orbits) $(G\times X)/H$.
If $T$ is a transversal for $G/H$ (that is, a set of coset representatives, exactly one per coset), then we can say every element of $G\times_HX$ is uniquely expressible as $(t,x)$ with $t\in T,x\in X$. So $G\times_HX$ is a disjoint union of $[G:H]$-many sets equinumerous with $X$ (indeed, they're all $G$-translates of the copy of $X$, namely $\{e\}\times X$, within $G\times_H X$).
This construction is the "most natural," since there is a universal property that stands out: for any $G$-set $Y$ containing $X$ there is a morphism $G\times_HX\to Y$ given by $(g,x)\mapsto gx$. That is, every $G$-orbit containing $X$ as an $H$-orbit is a homomorphic image of $G\times_H X$. More generally, you get Frobenius Reciprocity:
$$ \hom_H(X,A)\cong\hom_G(G\times_HX,A) $$
for any $G$-set $A$. That is, every $H$-intertwiner from $X$ into a $G$-set uniquely lifts to a $G$-intertwiner from $G\times_HX$. One may view $A$ as a $G$-set and (the restricted action) as an $H$-set as objects of different categories, in which case "restricting" actions is a functor, and we may write
$$ \hom_H(X,\mathrm{Res}_H^G A)\cong \hom_G(\mathrm{Ind}_H^GX,A). $$
The linear version of this works for linear group representations, in which case the explicit formula involves a tensor product, $\mathrm{Ind}_H^GV=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V$. Indeed, the same idea works for extension of scalars for modules, in particular complexification. (We may complexify a real vector space $V$ as $\mathbb{C}\otimes_{\mathbb{R}}V$, which we may view as $V\oplus iV$ so long as $V$ and $iV$ are understood to intersect trivially.)
