Is it permissible for my father to keep the money of my deceased maternal aunt—who gave him the money to trade with and share the profits, and told him: "If I live, it is my money, and if I die, it is yours, permissible for you"—or must the money be returned to her heirs?

The default rule is that the money left by your aunt is for her heirs. Anyone who claims she made a will in their favor must present valid Sharia proof unless the heirs confirm it. If the will is proven or the heirs confirm it, the legatee receives only one-third of the inheritance. Anything beyond that goes to the heirs, because a bequest to a non-heir is only valid up to one-third. Any amount exceeding one-third is not valid unless the heirs consent to it. Ibn Qudamah said: "A bequest to a non-heir is binding up to one-third without requiring permission. Anything beyond one-third depends on their (the heirs') permission. If they permit it, it is valid; if they reject it, it becomes void, according to the consensus of all scholars." The basis for this is the Prophet's (peace be upon him) saying to Sa'd: "A third, and a third is much," and his (peace be upon him) saying: "Indeed, Allah has bestowed upon you a charity from a third of your wealth, at the time of your death."