Can I work alone on an extension of a paper that had multiple authors? Is it permissible for one of the authors of a paper to work alone for the extension of that paper?
There is nothing illegal about this, nor is this publicly frowned upon.  After all, it's very possible for some of the original co-authors to simply not be interested in the extension.

However, your co-authors might see this as impolite, to say the least.  If you offer to include them in working on the extension and they decline, then everything is fine.  If you don't make such an offer to them, they might not want to work with you ever again.