Is it permissible for the husband to take back his wife whom he has revocably divorced – during her waiting period ('iddah) – given her bad character and communication with his brother, and the resulting doubts he has and his desire to avoid problems with his first wife?

Your wife's action of establishing a relationship with your brother during her 'iddah (waiting period) is a clear abomination. If she does not repent with sincere repentance, we do not advise you to take her back. It is recommended to divorce an unchaste wife or one who neglects the rights of Allah. You must consider the situation of your first wife and the potential problems when taking this woman back. As for your statement "I did not divorce her" while thinking of "returning to her after what I saw from her after the divorce," this is a contradiction. If the wife is in her 'iddah, she has rights to maintenance, clothing, and housing, and she is not permitted to expose herself to men. However, if she has exited her 'iddah, she has become irrevocably separated from you, and she has the right to marry whomever she wishes, and you are not permitted to keep her in suspense.