What is the ruling of Islam regarding a house that was in the aunt's name, then she gifted it to the son at the father's behest, knowing that the father is the true owner of the house and had disposed of some of his other properties during his lifetime?

If the house belonged to your father and he did not gift it to your aunt, then it is impermissible to designate one of the children with it unless each daughter's share is half the value of the house, or the daughters agree to it. Justice dictates that the male receives the equivalent of the share of two females, as in inheritance. If justice is not achieved, the distribution must be redone, and the gifts added to the estate and divided according to Islamic law. As for what the son spent on maintenance, if he donated it, it is not to be reimbursed; but if he spent it with the intention of being reimbursed, then he takes it from the estate. If the adult heirs agree on a specific division, then the matter is up to them. However, if the father gifted the house to your aunt and it became her property, and then she donated it to your brother, it becomes his property and does not enter into the father's estate. It is essential to maintain ties of kinship.