Fatefully is how you say fateful about an action or a way of happening. If someone fatefully agreed, that yes later cost them; the choice changed what came next, and not for the better. You will meet this word in books and news more than in chat. It is not an everyday classroom word. Same idea as fateful: a heavy turning point, usually with a bad shadow — only now it tells you how something was done, not what kind of noun it was.