In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
Answering your question, Dar Al-Ifta Al-Misriyyah, states:
It is disliked to talk during the sermon of the Friday prayer, but there is no harm in making istighfar (praying for forgiveness) between the two parts of the sermon. This does not invalidate the prayer as it is a Sunnah.
Almighty Allah knows best.