How to get forgiveness for a big mistake?

You should've told Francis "Thank you for keeping me in the loop, but I'm sure I will hear about it if it really happens." Then, instead of warning your very good friend about his girlfriend's behavior, you should've let him alone, he was going to find out anyway, and that put a lot of pressure onto him. You need to gather all three resources and give a sincere, face-to-face apology, and tell the couple that you're sorry for getting in between them. I would imagine that Francis is angry with you because you told your very good friend about his girlfriend apparently being unfaithful, and tell her then- "You kept me in the loop, I was only trying to protect (his) feelings, but I realize that I'm wrong now (even if you don't think you were wrong, still apologize, she's going to be bitter until you do), and I'm sorry." If they don't accept your apology, don't try to force them to, give them time. In time they will heal and forgive you. Just consider this a lesson learned, don't be too hard on yourself.