The flow of a Newtonian incompressible viscous fluid is a fundamental problem in computational sciences and engineering. In this case, the governing equations are the incompressible Navier--Stokes equations. For decades, researches have focused their interest on devising numerical methods for the solution of this problem. The non-linearity of the incompressible Navier--Stokes equations requires one to employ robust and efficient solvers for either the Picard or the Newton linearization of the discretized equations, which results in a sequence of linear systems to be solved in order to obtain a numerical solution. For instationary problems, the importance of a robust and efficient linear solver is even more evident, as at each time step one is required to solve a sequence of linear systems. In this talk, we consider the numerical integration of the instationary incompressible Navier--Stokes equations, when employing a Runge--Kutta method in time. The time discretization results in a non-linear system to be solved for the stages of the Runge--Kutta method at each time step. In order to find a numerical solution, we employ a Newton linearization of the non-linear problem, which is then discretized with suitable finite elements. The resulting linear systems present a saddle-point block structure, and can be very large and sparse in real-life applications. For this reason, in order to find a solution one requires the use of preconditioned iterative methods. We adopt an augmented Lagrangian-based preconditioner, and employ saddle-point theory for deriving approximations of the $(1,1)$-block and the Schur complement. Numerical experiments show the effectiveness and robustness of our approach, for a range of problem parameters and different Runge--Kutta methods.