Abstract: A countersinking-carrier nozzle has a flat front surface with a central hole via which emerges a countersinking drill, such that said flat front surface establishes a limit to the penetration depth of the countersinking nozzle. The nozzle includes a flexible retainer for the nozzle front surface so that it swivels when making contact with a curved countersinking surface, adapting itself tangentially to it while the headstock presses against the countersinking surface. The flexible retainer is defined by a ball-and-socket joint including a curvo-concave retaining receptacle, with the possibility of swivelling, with a complementary curvo-convex portion provided in the rear part of the flat front surface. This portion projects with respect to the receptacle to allow it to swivel when pressed against the countersinking surface, at which moment the complementary curvo-convex portion of the rear part of the front surface swivels in the receptacle of the headstock.