Stability is an important system property. In this paper we examine the steps toward fully automated stability verification for a class of discrete-time hybrid systems. These systems are hybrid in the sense that they can have several modes with possibly different discrete-time dynamics. For systems with affine discrete-time dynamics in each mode, we employ a method that is based on Lyapunov theory and reduces the verification task to convex optimization. We detail the steps that are needed for automatic stability verification using this method: 1) choice of a state space partitioning 2) calculation of possible transitions between the different regions of the partitioning and 3) conversion of the problem into linear matix inequalities, which can be solved through convex optimization. For steps 2) and 3) we present solutions that are suitable for full automation.