A generalized Brauer construction and linear source modules

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