From what I have read, Charlemagne had a separate crown (the one in my avatar) created for Pepin.
However, it was probably some web-based source like Wikipedia.
Nevertheless, the crown in my avatar exists and everywhere I see it, it is called the Crown of Pepin, King of Italy.
Thank you for the information about where the Italian crown jewels are (that's a pity, but I should have expected)
Edited to add:
Alas, I am a dull and overly optimistic at the same time. The crown in my avatar is a reconstruction of this one:
...one of the several crowns of Charlemagne, the one he gave to his second son, Pepin, and the crown went on to Bernhard, and I'm not even sure it still exists.
Hence, the attempt at reconstruction (by more than one person). Well, at least that means maybe I can get one!