What is the ruling of Islam regarding a husband taking back his wife after divorcing her with three pronouncements?

What appears to us from the question is that the three pronouncements of divorce were not interspersed with a revocation. Therefore, the fatwa (definition) is based on the opinion of the majority of scholars, which states that whoever divorces his wife three times, she becomes irrevocably divorced from him, and she is not lawful for him until she marries another husband, as Allah the Almighty says: {Divorce is twice. Then, either keep [her] in kindness or release [her] with good treatment. And it is not lawful for you to take anything of what you have given them, unless both fear that they will not be able to establish the limits of Allah. But if you fear that they will not establish the limits of Allah, then there is no blame upon either of them concerning that by which she ransoms herself. These are the limits of Allah, so do not transgress them. And whoever transgresses the limits of Allah - it is those who are the wrongdoers.}