Hi all,
Thank you very much for your support.
The problem is fixed! ;D

I have found another topic with the same problem and an hint to Microsoft One Drive, formally known as SkyDrive.
Unfortunately I had forgotten that the PC with the Problem uses an Online-Microsoft User Account. That’s the difference to the other PC’s.

I deactivated the start of OneDrive by using gpedit.msc
… and after rebooting the system the avast icon was in the tray.

Everything is fine now.

Thanks once again

Nullquadrat