@Old_Grey You get a .fz file (and more) by unzipping a .fzz file. It is the uncompressed sketch file. The “and more” would be the part files (fzp and svg) that belong with the sketch that are not part of core. So if someone unziped a .fzz (or just saved in uncompressed mode), then posted the .fz, the (non-core) part files would all be missing.
@StefanL38 Is the content of the problem fzz private? Can you post it here? (7th button from the left when editing a post or comment) With the actual sketch file, I can unpack it manually, to see what it contains, and what it tries to reference. The description so far says that for some reason Fritzing did not include that part in the sketch file. Do not put the fritzing program folder in Documents/Fritzing. That is intended to be the user data area. Putting the code there could end up mixing the core parts and user parts. The symptoms described do NOT sound like an installation problem. It could be a bug in the program that your specific sketch triggers. If you can upload both the sketch document, and the fzpz for the part that is missing when your friend opens it, I can look at both what parts are currently in the sketch, and try to recreate it, and check again if the part gets included.
It could be a problem with the part file itself. The fzpz files are created by various people, with varying quality. Some look OK, but are broken in different ways. I have some tools to check that too, if you can post the part file. Maybe the part file is good enough to open and use, but broken in a way that Fritzing can not save it in the .fzz file. But works for you, since it is found in the folder mentioned in that error message. Speculation. I need the files to try to verify.