What is the ruling on a woman praying in the mosque under these circumstances, and do you advise me to accept an offer to go to Friday prayer at a distant mosque with three girls and a male colleague driving the car?

The purified Sunnah indicates that the most concealed place, farthest from the mixing of men, is the best for a woman's prayer, and the best prayer for a woman is in her home. The Hadith explicitly states that it is better for a woman to pray in her home and not go to the mosque for the five daily prayers or Jumu'ah, even if it is permissible for her to do so. A woman traveling to a distant mosque to perform Jumu'ah without a mahram is not correct, because Jumu'ah is not obligatory for her, and traveling without a mahram is forbidden for her, unless the mosque is not at a traveling distance. If she benefits from going to the mosque by learning rulings of her religion that she cannot learn except by going there, and this is not at a traveling distance, then it is preferable for her to go, even if it is not obligatory.