Modularity of certain mod p^n Galois representations

For a rational prime $p \geq 3$ and an integer $n \geq 2$, we study the modularity of
continuous $2$-dimensional mod $p^n$ Galois representations of
$\Gal(\overline{\Q}/\Q)$ whose residual representations are odd and absolutely
irreducible. Under suitable hypotheses on the local structure of these representations
and the size of their images we use deformation theory to construct characteristic $0$
lifts. We then invoke modularity lifting results to prove that these lifts are modular. As
an application, we show that certain unramified mod $p^n$ Galois representations
arise from modular forms of weight $p^{n-1}(p-1)+1$.