I was wondering if Alizée has ever made a Christmas song, so... Apparently yes
(She starts singing at 0:53)
Unfortunately it is not just her, but at least the title of the video made the research easier. The song is nice.
I think Alizée is the kind of person that would be perfect to sing a Christmas song: she is sunny, kind, friendly with kids and her voice is delicate and comforting. It would be nice if she made a new song and became as those singers whose songs are everywhere during this period.