Why don't airlines also do charter jets?

This evening, I watched a documentary on TV about the rise of private jets.
During it, they talked about the rise of charter jet companies. These being
companies from whom you can charter a private jet from a one-time use to
something along the lines of a time-share. Anyway, what struck me odd was
that no airline company ... not even small ones ... are in the private jet
charter business. I've seen other TV programs about private jets or where
they're at least talked about for a bit and none of these ever mentioned
airlines getting into the charter jet business ... or even why they're not
in them. Why aren't they?

To me it would seem to be a logical side division for the airlines to get
into. I'd just treat it as one step up from first class seats. Coach
Business First Class Private Jet. Additionally, they could possibly
take advantage of the quantity discount they get on jet fuel for these
private jets.

Or are they currently in this business and the programs I've been catching
either didn't think it was important enough to mention or didn't know this?

Or are the airlines prohibited from going into the charter jet business by
the FAA?

Or is charter jets nothing like airlines so there could be no synergy
savings?

Or is the charter jet business so much of a roller-coaster type of a
business sector that it would lose the airlines money?

If no airline currently is in the charter jet business, have they ever even
given it a try? If so, I'd enjoy hearing how it went and why they left it.

Thanks in advance!

Scott Jensen
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