Is a friend obligated to flog his friend who committed adultery, especially since his friend insists on having the Hadd punishment carried out on him as a deterrent from returning to the sin, knowing that carrying out the Hadd is an expiation for the sin?

Punishments (Hudud) are only to be carried out by the ruler or his deputy. It is obligatory upon whoever commits adultery to conceal themselves with Allah's concealment, to be sincere in their repentance, to resolve to abandon it, and to frequently seek forgiveness and perform righteous deeds. This is based on the Almighty's saying: "And those who, when they commit an indecency or wrong themselves [with sin], remember Allah and ask forgiveness for their sins - and who can forgive sins except Allah? - and [who] do not knowingly persist in what they have done," and His saying: "But whoever repents after his wrongdoing and reforms, indeed, Allah will turn in forgiveness to him. Indeed, Allah is Forgiving and Merciful." He should also strive for marriage, observe voluntary fasting, attend gatherings of goodness, frequently supplicate and remember Allah, lower his gaze, and avoid mixing with non-mahram women and bad company.