The phrase "head the house" refers to being in charge of the household and making important decisions that affect the family. There are several synonyms that can be used instead of this phrase, including "run the household", "manage the home", "oversee the family", "lead the household", and "take charge of the house". These synonyms convey similar meanings and are useful in expressing different shades of meaning. For example, "oversee the family" suggests a higher level of authority and responsibility than "run the household". Ultimately, whatever word or phrase is chosen to describe this role, it requires diligence, patience, and a strong sense of leadership.