To what extent is the hadith sound which mentions that Aisha held a mourning lamentation for Abu Bakr, and that Umar ibn al-Khattab forbade them from doing so, and when they did not comply, he ordered Umm Farwah, Abu Bakr's sister, to be brought out and struck with a whip?

This narration was reported by Al-Tabari, Ibn Sa'd, Ibn Al-Athir, and Al-Hindi with the same wording. Ibn Shabbalso reported it in his "Tarikh Al-Madinah" with the aforementioned wording, on the authority of Al-Zuhri from Sa'id ibn Al-Musayyab. Al-Bukhari reported the core of the story as a suspended narration in his Sahih. Ibn Hajar connected the story with a sound chain of narration, through Al-Zuhri from Sa'id ibn Al-Musayyab, mentioning that Umar prohibited wailing over Abu Bakr. When Aisha and those with her refused to stop, he ordered Hisham ibn Al-Walid to strike Umm Farwah, Abu Bakr's sister, with a whip to disperse the wailers. Ibn Al-Jawzi clarified that Umar disciplined Umm Farwah out of respect for Aisha. The chain of narration for the story up to Sa'id ibn Al-Musayyab is sound and connected, but it is Mursal (lacks a direct link to the Prophet) and disconnected concerning those above Sa'id, because he was born two or four years after the incident during Umar's caliphate.