Modular supercuspidal lifts of weight $2$

We prove that for a non CM Hecke cuspform $f\in S_k(\Gamma_0(N))$ and a prime $p>\max\{k+1,6\}$ and $p\nmid N$ such that the residual $p$-adic Deligne representation $\overline{\rho}_f$ is absolutely irreducible and $\mathrm{SL}_2(\mathbb{F}_p)\subseteq \mathrm{Im}(\overline{\rho}_f)$, there exists a modular supercuspidal lift $\rho_g$ with $g\in S_2(Np^2,\epsilon)$ for some Nebentypus character $\epsilon$. We apply this result to correct a mistake in \cite{dieulefait}, where the micro good dihedral prime $p=43$ is introduced to prove the automorphy of $\mathrm{Sym}^5$ of level $1$ modular forms. We also discuss how this result is versatile enough to prove other instances of Langlands functoriality like, for instance, in the safe chains introduced in \cite{luissara} and \cite{GL2GL2GL2}, where the automorphy of tensor products of certain modular or automorphic representations is established