Traditional vertical handover schemes postulate that vertical handovers (VHOs) of users come on an individual basis. This enables users to know previously the decision already made by other users, and then the choice will be accordingly made. However, in case of group mobility, almost all VHO decisions of all users, in a given group (e.g., passengers on board a bus or a train equipped with smart phones or laptops), will be made at the same time. This concept is called group vertical handover (GVHO). When all VHO decisions of a large number of users are made at the same time, the system performance may degrade and network congestion may occur. In this paper, we propose two fully decentralized algorithms for network access selection, and that is based on the concept of congestion game to resolve the problem of network congestion in group mobility scenarios. Two learning algorithms, dubbed Sastry Algorithm and Q-Learning Algorithm, are envisioned. Each one of these algorithms helps mobile users in a group to reach the nash equilibrium in a stochastic environment. The nash equilibrium represents a fair and efficient solution according to which each mobile user is connected to a single network and has no intention to change his decision to improve his throughput. This shall help resolve the problem of network congestion caused by GVHO. Simulation results validate the proposed algorithms and show their efficiency in achieving convergence, even at a slower pace. To achieve fast convergence, we also propose a heuristic method inspired from simulated annealing and incorporated in a hybrid learning algorithm to speed up convergence time and maintain efficient solutions. The simulation results also show the adaptability of our hybrid algorithm with decreasing step size-simulated annealing (DSS-SA) for high mobility group scenario.