Is transmitting slanderous speech against a woman with the intention of informing her about it considered forbidden backbiting, knowing that this speech caused a problem and that the unfortunate person is believed to be the source of the rumor?

The questioner should have defended his sister's honor and forbidden the speaker from backbiting her. If he fell short, he would be sinful. Evidence for this includes:

1. The hadith: "Whoever defends the honor of his brother when he is backbitten, it is a right upon Allah to free him from the Fire."

2. The hadith: "No man abandons a Muslim man in a place where his sanctity is violated and his honor is disparaged, except that Allah عز وجل abandons him in a place where he loves His support."

As for conveying the news to the sister without mentioning the person's name or anything that identifies him, it does not fall under prohibited tale-bearing (namima), but it should not be done, as its harm outweighs its benefit.