Under U.S. copyright law, yes. A sequel is a derivative work, i.e. a work based on or derived from an underlying work. Only the copyright holder of the original work may authorize derivative works.
Copying text verbatim is copyrigt violation. Plagiarism (copying ideas) is a different issue and much more of a gray zone. In general ideas are not protected, but it can be prosecuted in blatant cases. The described case seem to be a bit of both.
Right, but what are the chances that the "new material" doesn't include any elements of protected expression from the underlying work, e.g. characters? I understood the request to be about writing a sequel.
I'm not in the US, so I could easily be off base.
I really enjoyed The Time Ships by Stephen Baxter, which is a sequel to The Time Machine by H. G. Wells. He's a UK author, so maybe it's different.
As a society I think we are better off with this kind of remixing so I'd hope that's the direction we head in.