A continuator is a writer who chooses to finish the work of another author.
(The latest book to be “continuated” is Charles Dickens’ unfinished novel, The Mystery of Edwin Drood, his only mystery story. It was his final book, published in 1870, but he only completed six of twelve parts before he died. No-one knows how he intended to conclude it, but Gwyneth Hughes not only adapted Dickens’s half-written novel into a new television drama but also gave it an ending of her own. You could describe what continuators do as literary necrophilia, or maybe they simply keep popular authors alive.)