Two-point extensions over irrational rotations given on $\mathbb {T} \times \mathbb {Z}/ 2\mathbb {Z}$ by $T_\beta (x,y)= (x + \alpha \hbox { mod } 1, y + \mathbf {1}_{[0,\beta [}(x) \hbox { mod }2)$, where $\beta \in {}]0,1[$ and $\alpha $ is irrational have been extensively studied from the spectral point of view, starting with a well known paper by A. Katok and A. Stepin. In particular, they always have simple spectrum as soon as they are ergodic, and the question of the existence of such a transformation whose spectral type admits an absolutely continuous part arises naturally. We give here a negative answer to this question, by showing for every $\alpha $ and $\beta \in {}]0,1[$ the spectral singularity of these 2-point extensions.