Lifting residual Galois representations with the same semi-simplification

We consider two modulo $p$ representations $\overline{ρ_1}:Γ_F \to G_1(k)$ and $\overline{ρ_2}:Γ_F \to G_2(k)$, for $F$ a totally real field, and $G_1$ and $G_2$ possibly distinct reductive groups. We show that under certain conditions both representations admit geometric $p$-adic lifts by allowing