A generalized Brauer construction and linear source modules

For a complete discrete valuation ring O \mathcal {O} with residue field F F , a subgroup H H of a finite group G G and a homomorphism φ : H → O × \varphi : H \to \mathcal {O}^\times , we define a functor V ↦ V ¯ ¯ ( H , φ ) V \mapsto \overline {\overline {V}} (H,\varphi ) from the category of O G \mathcal {O} G -modules to the category of F N G ( H , φ ) FN_G(H,\varphi ) -modules and investigate its behaviour with respect to linear source modules.