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.
I once was in the situation that my coauthors were always "interested" in the extensions, but had little time, which delayed the whole paper.
After some time I began to write extensions on my own and tried to publish them, which, indeed, was bad for the relationship to my coauthors. They had the impression that I was working behind their back (which was somehow true) - while my impression was that I was much quicker alone.
Are you (writing the papers) under someone's employ or as part of a contract with someone? Then supposedly you would be expected to make sure your employer / contractor is OK with whatever direction you go. If you are not, then you simply have no such obligations anymore.