What is the ruling on swearing falsely on the Qur'an with a double meaning (tawriyah) to appease one's wife, and what is its expiation?

It is not permissible for a wife to spy on her husband. If the husband uses an equivocation in his oath and does not lie, there is no sin upon him. The decisive factor is the intention of the swearing husband, not the intention of the wife who requests the oath. However, if the husband lies and does not use equivocation, then he must repent to Allah Almighty.