  We extend the lifting methods of our previous paper to lift reducible odd
representations $\bar{\rho}:\mathrm{Gal}(\overline{F}/F) \to G(k)$ of Galois
groups of global fields $F$ valued in Chevalley groups $G(k)$. Lifting results,
when combined with automorphy lifting results pioneered by Wiles in the number
field case and the results on the global Langlands correspondence proved by
Drinfeld and L. Lafforgue in the function field case, give the only known
method to access modularity of mod $p$ Galois representations in both reducible
and irreducible cases. In the reducible case this allows one to show that the
actual representation, rather than just its semisimplification, arises from
reduction of the geometric representation attached to a cuspidal automorphic
representation on the dual group of $G$. As a particularly concrete
application, we get a version of Serre's modularity conjecture for reducible,
odd representations $\bar{\rho}: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})
\to \mathrm{GL}_2(k)$. This extends earlier results of Hamblen and Ramakrishna
in this classical case and proves modularity of infinitely many extensions of
fixed characters that are not covered by loc. cit.
