Let $f$ be a cuspidal Hecke Eigenform of weight $k \geq 2$ and let $\rho_{f, \lambda}:G_{\mathbb{Q}} \rightarrow GL_2(E_{\lambda})$ be the corresponding Galois representation with $2 \mid \lambda$ constructed by Deligne. Assume now that $\pi_f={\otimes}' \pi_p$ be the corresponding automorphic representation.

My question is the following: If we assume $\pi_2$ non-monomial (it is not induced by character), can you say that projective image $\tilde{\rho_{f, \lambda}}:G_{\mathbb{Q}} \rightarrow PGL_2(E_{\lambda})$ is an exceptional group (S_4). I am aware the image of the corresponding Weil-Deligne representation is $S_4$.