  This paper studies sequential quantum games under the assumption that the
moves of the players are drawn from groups and not just plain sets. The extra
group structure makes possible to easily derive some very general results
characterizing this class of games. The main conclusion of this paper is that
the specific rules of a game are absolutely critical. The slightest variation
in the rules may have important impact on the outcome of the game. This work
demonstrates that it is the combination of two factors that determines who
wins: (i) the sets of admissible moves for each player, and (ii) the order of
moves, i.e., whether the same player makes the first and the last move. Quantum
strategies do not a priori prevail over classical strategies. By carefully
designing the rules of the game it is equally feasible either to guarantee the
fairness of the game, or to give the advantage to either player.
