What is the correct Islamic legal action to be taken towards a wife who has confessed to adultery with a foreign man, and is what occurred considered actual zina (fornication/adultery), taking into account the presence of children and a pregnancy, and the discovery of similar behaviors among some of her female relatives?

It is not permissible to spy on one's wife or to seek out her faults unless there is a valid reason, such as the emergence of suspicion from her. Advise your wife to repent, and if her repentance is sincere, then keep her honorably and conceal her faults. Prevent her from visiting her family if you fear they will corrupt her. If she returns to forbidden relationships, then divorce her. The husband must fulfill his right of qawamah over his wife, close the doors to fitna for her, establish God's limits in his home, and cooperate with her in obeying God.