Lifting residual Galois representations with the same semi-simplification

If a $p$-adic Galois representation ${\rho }_{f,\nu }:{\mathrm{\Gamma }}_{\mathbb{Q}}\to {\text{GL}}_2(E_{f,\nu })$ attached to some eigenform $f$ is residually reducible it will have 2 non-isomorphic reductions, which have the same semi-simplification. In this paper, we answer a version of the inverse question, first brought up by Toby Gee and Alice Pozzi. Starting with two modulo $p$ non-semi-simple representations $\overline{{\rho }_1},\overline{{\rho }_2}:{\mathrm{\Gamma }}_{\mathbb{Q}}\to {\text{GL}}_2(k)$, which have the same semi-simplification we show that under some mild conditions that they are reductions of representations attached to newforms of the same weight $r\ge 2$, the same level $N\ge 1$, and the same Neben character $\epsilon$.