Suppose we have a simultaneous game, that has a strong Nash equilibrium (SNA), i.e. a weak Pareto efficient Nash equilibrium (no deviation of any subset of player brings a benefit to them).

Now suppose we play this game repeatedly. Does the repeated game has a strong Nash equilibrium, too?

I would keep the question about the payoff function of the repeated game as open. Choose whatever adopted payoff works for the answer.

The idea behind the question is as follows: Games that don't have SNA, like prisoner's dilemma, might have those in the repeated scenario, since additional effects like long term strategies come into play.

Based on this, my guess is, that playing the SNA in every game, would also give a SNA in the repeated game.