Is it permissible for a husband to take back his wife if he divorced her with a single pronouncement but told her family it was three pronouncements for a benefit he perceived?

It is not clear whether the divorce actually occurred or if it was merely an agreement. A mere agreement to divorce does not constitute a divorce, because it is a promise to divorce and not an actual enactment of it. The husband's statement "I divorced her" made as a false declaration is considered a divorce by the Shafi'is and Hanafis judicially, but not religiously, whereas the Hanbalis consider it a divorce both judicially and religiously. Unless a Sharia judge rules it as such, this declaration has no effect. What is written in official divorce documents is not a judicial ruling. If the husband actually divorced his wife three times, the divorce is effective according to the majority of scholars. If the matter was limited to an agreement without actual enactment, she remains his wife. If the Sharia judge rules for a triple divorce based on what he uttered, the divorce is effective, and she is not lawful for him until she marries another husband. Moral compulsion does not prevent the occurrence of divorce; only absolute coercion does.