Fatimah was deeply annoyed with 'A'ishah, since the women of Medina reported to her the offending words of A'ishah.

Ibn Abi al-Hadid

Next to the remarks of Imam 'Ali about 'A'ishah's long-standing rancor towards him, we now hear the words of Ibn Abi al-Hadid about this matter. He says: When. I was studying theology, I read out this sermon of the Imam in the presence of my master, ash-Shaykh Abi Ya'qub Yusuf, son of Isma'il al-Lama'ani (may God save him), and asked him to explain the philosophy behind Imam's utterance. He complied with my request and gave an elaborate explanation, whose summary I quote here. I cannot remember all his words, so I must confine myself to this brief account. A part of it comprises his own words, while the rest is mine based on his own ideas. Shaykh Abu Ya'qub says:

A step-mother for Fatimah
The hostility between 'A'ishah and Fatimah began when the Prophet married A'ishah upon Khadijah's death, and she took her place. It was natural for Fatimah to be dissatisfied at having a stepmother, and also natural for a woman to be vexed at her husband's affection for this daughter of his former wife and for the second wife too.
Similarly a daughter dislikes her father's attention to another woman who is a rival of her mother, even though she is no longer alive. Even if Khadijah were still alive and 'A'ishah entered the Prophet's house, their quarrel would be more noisy and violent. Now that she was dead, this hostility would be transferred to and inherited by her daughter.
Moreover, it is said that the Prophet loved 'A'ishah very much(81) and showed her favour. So the more the Prophet showed affection to the new wife, the more uneasy became Fatimah and was more deeply hurt.