Modularity of $operatorname{PGL}_2(mathbb{F}_p)$-representations over totally real fields

We study an analogue of Serres modularity conjecture for projective representations $overline{rho}: operatorname{Gal}(overline{K} / K) rightarrow operatorname{PGL}_2(k)$, where $K$ is a totally real number field. We prove new cases of this conjecture when $k = mathbb{F}_5$ by using the automorphy lifting theorems over CM fields established in previous work of the authors.