Permutation homomorphisms lift through modular reduction

For coset permutation modules, Mackey restriction formula and Frobenius reciprocity identify the homomorphism space with a free module having one orbit-sum basis vector per double coset. The same basis works over the residue field, so reduction of homomorphisms is surjective. Applied to endomorphisms of a permutation module, idempotent refinement theorem lifts each modular direct summand to a trivial source lattice. Lifting inverse isomorphisms between reductions shows uniqueness among trivial source lattices.