Should one prostrate for the recitation (of the Quran) with a group that is reciting the Quran before the Friday prayer, if one believes this gathering to be an innovation?

The prostration of recitation (Sajdat al-Tilawah) is a Sunnah (recommended) and not obligatory. It is prescribed for the one who listens attentively, not for the one who merely hears unintentionally. It has been narrated from 'Uthman, Ibn 'Abbas, and 'Imran that the prostration is for the one who listens attentively. Therefore, if you are listening to a group reciting the Qur'an, it is prescribed for you to prostrate with them when they pass by an verse of prostration.