Let $R$ be a commutative ring, like the ring of integers $\mathbb Z$ or the ring of $p$-adic integers $\mathbb Z_p$. Let $G$ be a finite group; let us consider permutational representations of $G$ over $R$, i.e., $R[G]$-modules of the form $R[G/H]$, where $H\subset G$ is a subgroup, and direct sums of such modules. These are free $R$-modules where $G$ acts so that there exists a basis of the module preserved (as a whole) by the action.
I am interested in finite exact sequences of representations of the above type, particularly in those of them that are not very long. There are some beautiful examples, e.g., for $G=\mathbb Z/2$ and $R=\mathbb Z/2$ there is an exact sequence $$ 0\rightarrow R\rightarrow R[G]\rightarrow R\rightarrow 0, $$ while for $G=\mathbb Z/n$ and $R=\mathbb Z$ there is an exact sequence $$ 0\rightarrow R\rightarrow R[G]\rightarrow R[G]\rightarrow R\rightarrow 0. $$ For the fourth symmetric group $G=\mathbb S_4$ and $R=\mathbb Z$ there is an exact sequence $$ 0\rightarrow R\rightarrow R[\mathbb X_4]\oplus R\rightarrow R[\mathbb X_6]\oplus R\rightarrow R[\mathbb X_3]\rightarrow0, $$ where $\mathbb X_4$ is the four-element set that $\mathbb S_4$ permutes, $\mathbb X_6$ is the set of all two-element subsets of $\mathbb X_4$, and $\mathbb X_3$ is the quotient set of $\mathbb X_6$ by the obvious involution. Dihedral groups also have some four-term exact sequences of permutational representations.
Where is one supposed to get such exact sequences? There are some obvious ways, like e.g. one can take cones of morphisms of exact sequences of this type, or one can do restriction or induction from one group to another one. Are there any other constructions?
For constructions to be interesting, they should of course be removed far enough from the trivial case when $|G|$ is invertible in $R$. E.g., to have $R=\mathbb Z_p$ and $|G|$ a large $p$-group would be perhaps most highly nontrivial.
EDIT: One of the commenters asked where does the sequence for $\mathbb S_4$ come from, so let me say a few words about this. Not that I really understand it, but there is a geometric construction using a CW complex, and not quite of the kind that Greg suggests in his second answer below.
Represent the group $\mathbb S_4$ as the group of rotations of the $3$-dimensional cube. Consider the quotient CW complex of the cube's surface by the central symmetry involution. The group $\mathbb S_4$ still acts on the quotient. The set of vertices of the quotient is the $\mathbb S_4$-set $\mathbb X_4$, the set of edges is the $\mathbb S_4$-set $\mathbb X_6$, and the set of faces is the $\mathbb S_4$-set $\mathbb X_3$.
Now consider the map $R[\mathbb X_4]\rightarrow R[\mathbb X_6]$ assigning to a vertex the sum of the three edges ending in it (without any signs!) and also the map $R[\mathbb X_6]\rightarrow R[\mathbb X_3]$ assigning to an edge the sum of the two faces bordering on it (also without any signs!). The composition of these two maps is not zero, of course; what it does is taking every vertex to twice the sum of all the three faces. One somehow transforms this pair of arrows into a four-term exact sequence by adding the trivial $\mathbb S_4$-module direct summands $R$ in several degrees.