If $R$ is a field, then one approach that explains at least some constructions is to extend a permutation module $P$ to a chain complex with semisimple homology, then resolve irreducible modules with permutation modules. The second half, permutation resolutions of irreducible modules, then becomes a modified question that can be studied. Sometimes you can place permutation modules in an exact sequence of more general modules, and then resolve those other modules by permutation modules in some convenient way.

For instance in characteristic zero, many groups (or all of them?) have enough permutation modules to express every irrep as a formal linear combination of them; and any such formal linear combination can be dressed up to a resolution. In characteristic 0, the question reduces to linear dependences among characters of permutation modules, which is a lame answer but at least valid for part of the question.

On that theme, here is a preprint by Boltje and Hartmann that constructs a conjectured resolution of Specht modules of $S_n$ (over $\mathbb{Z}$) by Young modules. This is presumably a related tool.

Here is a preprint by Boltje and Hartmann that constructs a conjectured resolution of Specht modules of $S_n$ (over $\mathbb{Z}$) by Young modules. This is presumably a related tool.

(An earlier version of this answer had some out-of-step comments about resolutions that were either not helpful or already addressed in the original question.)

If $R$ is a field, then one approach that explains at least some constructions is to extend a permutation module $P$ to a chain complex with semisimple homology, then resolve irreducible modules with permutation modules. The second half, permutation resolutions of irreducible modules, then becomes a modified question that can be studied.

For instance in characteristic zero, many groups (or all of them?) have enough permutation modules to express every irrep as a formal linear combination of them; and any such formal linear combination can be dressed up to a resolution. In characteristic 0, the question reduces to linear dependences among characters of permutation modules, which is a lame answer but at least valid for part of the question.

On that theme, here is a preprint by Boltje and Hartmann that constructs a conjectured resolution of Specht modules of $S_n$ (over $\mathbb{Z}$) by Young modules. This is presumably a related tool.

If $R$ is a field, then one approach that explains at least some constructions is to extend a permutation module $P$ to a chain complex with semisimple homology, then resolve irreducible modules with permutation modules. The second half, permutation resolutions of irreducible modules, then becomes a modified question that can be studied. Sometimes you can place permutation modules in an exact sequence of more general modules, and then resolve those other modules by permutation modules in some convenient way.

For instance in characteristic zero, many groups (or all of them?) have enough permutation modules to express every irrep as a formal linear combination of them; and any such formal linear combination can be dressed up to a resolution. In characteristic 0, the question reduces to linear dependences among characters of permutation modules, which is a lame answer but at least valid for part of the question.

On that theme, here is a preprint by Boltje and Hartmann that constructs a conjectured resolution of Specht modules of $S_n$ (over $\mathbb{Z}$) by Young modules. This is presumably a related tool.