  Del Pezzo fibrations appear as minimal models of rationally connected
varieties. The rationality of smooth del Pezzo fibrations is a well studied
question but smooth fibrations are not dense in moduli. Little is known about
the rationality of the singular models. We prove birational rigidity, hence
non-rationality, of del Pezzo fibrations with simple non-Gorenstein
singularities satisfying the famous $K^2$-condition. We then apply this result
to study embeddings of $\operatorname{PSL}_2(7)$ into the Cremona group.
