Is it permissible to transfer ownership of an apartment built in the father's house to one of the siblings for the sake of Allah the Almighty, without any compensation?

The father's mere permission to his son to build on top of his house, without explicitly stating it as a gift (hiba), does not make it a valid gift. Rather, it is a loan (ariya) that terminates upon the father's death, and the son is entitled to the cost of construction. However, if the father explicitly declared it a gift and had witnesses to it, then the apartment is the son's property, and he has the right to relinquish it to whomever he wishes among his siblings. Equity is preferable. The questioner's feeling of injustice is valid if it was a legal gift and the father does not have the means to equalize among his children. The preponderant opinion is the obligation to equalize among children in giving.