The problem:

Crysis 1 and Warhead have the bug that when playing in full screen mode, the lowest refresh rate of the monitor is used. So in my case that's 50HZ. But of course it should be 60 or more.

The solution:

First you download the Borderless Gaming tool.
https://github.com/Codeusa/Borderless-Gaming/releases

The tool is also available here on Steam, so if you want to support the developer you can buy it for a few euros / dollars.

1 - Start the tool

2 - Now you start Crysis and activate the window mode.

3 - now switch to the desktop with Alt Tabulator and press the maximize button. This is the third button from the top in the middle of the two windows.
https://gtm.steamproxy.vip/ugc/1833521630888883444/F34A3BAF5955941AF71180B0EFD4CFDDB4F96922/?imw=256&&ima=fit&impolicy=Letterbox&imcolor=%23000000&letterbox=false
Now you add the game to your favorites and in the future this process will be fully automatic.
Of course, only as long as you have the tool running in the background. I recommend starting it with windows.

4 - Play Crysis with the maximum FPS that your hardware actually allows.