In this machine the displacement assembly allows the working head to be moved in five or six axes, namely three translational movements along the three Cartesian axes X, Y and Z, and two or three pivoting options; the working head includes several tools for performing the various operations involved in fitting a fastener, particularly a drilling tool, a countersinking tool; these tools can each be moved in turn between a rest position and working position situated on the working axis; in this working position, each tool can move translationally along the working axis; the working head also includes a bearing piece mounted on it intended to come to bear against the components that are to be assembled and including a buffer region with which the drilling and countersinking tools come into contact as they move, this contact determining the fully deployed position of these tools.