What is the correct Islamic legal action regarding the father's share of the inheritance, in the absence of witnesses or clear evidence of his waiver of it to one of his aunts, and the existence of an oath from another aunt concerning the waiver's inclusion of other aunts, knowing that the father passed away before being aware of the intention to build a Quran memorization house?

If there is no evidence that your father gifted his share to your paternal aunt, then the claimant of this gift must provide legal proof. Otherwise, you must swear an oath that you have no knowledge of it, for the rule is: "The burden of proof is on the claimant, and the oath is on the one who denies." And even if there were evidence of the gift, the donee does not own the gift except by taking possession of it during the lifetime of the donor. Otherwise, it is not enforceable, and your father's share becomes an inheritance to be divided among his heirs.