Is it an act of trustworthiness to inform the family of my friend—who object to her marriage to an unsuitable person—that she is still involved with him, or is this considered instigation, given that I know she told them she left him and refuses to discuss the matter?

If the family’s rejection of a man is not based on his lack of religiosity or character, then the family has the right to reject him. It is permissible to secretly inform them of her attachment to him if all means of convincing her to leave him have been exhausted, due to the overriding benefit in preventing immorality or secret marriage. However, if the rejection is not based on a legitimate Shar'i impediment, then the advice should be directed to the family not to prevent the marriage, otherwise their prevention would lead to widespread tribulation and corruption.