Is there a natural permutation representation of a wreath product of groups? Is there a "natural" embedding of a $G \wr H$ into the group of permutation matrices? 
Like an element of $G\wr H$ looks like, $g=((g_1,g_2,..,g_{\vert H \vert}),h), \forall g_i \in G, h \in H$. Now one can imagine a vector space in standard basis as $\{ e_1,e_2,..,e_{\vert H\vert \vert G \vert}  \}$. I would like to know how would $g$ act on this basis? (I would believe that a natural action exists!)
If the above exists then anyone knows what its irreducible components look like? 
 A: When wreath products were initially introduced, they were defined as the wreath product of two permutation groups. A permutation group is by definition a pair $(G,X)$ of a group $G$ and a $G$-set $X$. And often such a permutation group was denoted $G$, for historical reasons going back to Galois, which is often confusing nowadays since a group generally means an abstract group with no permutation representation attached: of course any group has a canonical permutation representation attached, namely by action by left translations, but this is especially confusing when it has another canonical permutation representation, e.g. when we consider the permutation group $\mathfrak{S}_n$.
If $(G,X)$ and $(H,Y)$ are permutation groups, then the permutational wreath product $H\wr_X G=H^X\rtimes G$ acts on $Y\times X$ in the obvious way: if $h\in H^X, g\in G$, $(y,x)\in Y\times X$, then $(h,g)(y,x)$ is defined to be equal to $(h(g.x).y,g.x)$; we thus define $(H,Y)\wr (G,X)=(H\wr_X G,Y\times X)$ and call it the wreath product of these two permutation groups. Taking wreath products of permutation groups is naturally associative.
Now if $G,H$ are just groups, they define permutation groups in at least one canonical way, namely endowed with the left action on themselves, and the wreath product is then the standard wreath product $H\wr G$ (but not endowed with the left action on itself!). Beware that in contrast, taking standard wreath products of groups is not associative in any reasonable sense (for finite groups the cardinals do not always match).
Here I answered how you represent the group as permutations. How (finite) permutation groups $\mathfrak{S}_n$ are canonically represented as $n\times n$ matrices is obvious; how this decomposes into irreducibles under the action of some given permutation group is much more complicated and I don't think I can say many general things about this.
