Answer by Monthly Al-Sunnah, Jhelum
The Imam and the congregation should recite Ta‘awwudh (أعوذ بالله من الشيطان الرجيم) silently in prayer. There is no authentic evidence proving that the Prophet Muhammad (ﷺ) recited Ta‘awwudh aloud. The Companions (رضي الله عنهم) also used to recite it silently.
Evidence from the Companions
Saeed bin Mansur (رحمه الله) narrated in his Sunan:
"Abu Wail Shaqeeq bin Salamah (رحمه الله) said: The Companions (رضي الله عنهم) used to recite ‘Bismillah’ and ‘Ta‘awwudh’ silently in prayer."
[Nasb al-Rayah by Al-Zayla‘i: 1/358, Al-Dirayah by Ibn Hajar: 1/135 – Authentic Chain]
General Principle in Salah
The default rule in Salah is to recite silently, except in those instances where there is explicit evidence for reciting aloud.
Reciting Bismillah aloud is proven in some cases, but there is no authentic proof of reciting Ta‘awwudh aloud.
Conclusion
Both the Imam and the congregation should recite Ta‘awwudh silently before reciting Surah Al-Fatihah.
