You edited your post so mine just looks confused
First of all don't edit /etc/X11 Xmodmap it can get confusing, and at least you'll always know roots config is right.
Add one to your home directory start it with a period. so its ~/.Xmodmap
This is what I did for My MS office keyboard.
Now you need to execute the command xmodmap ~/.Xmodmap
and voila your sorted.
To execute everytime add the above command to .bashrc or similar.
Now in KDE goto the control centre Regional & Acessibility, your looking for Keyboard shortcuts.
In Gnome Desktop Prefrences Keyboard Shortcuts to assign your new F22 Key.