After starting the engine water is pushed out of the radiator?

It sounds like the thermostat is not opening completely, or they missed the head gasket failure which happens because the chemical test is only about 80% correct. Are you losing coolant? Let's try the thermostat first.

Here is a guide that will help walk you through the steps with instructions below for your car:

https://www.2carpros.com/articles/replace-thermostat

This guide can help as well:

https://www.2carpros.com/articles/head-gasket-blown-test

Check out the images (below). Let us know what happens and please upload pictures or videos of the problem.

Okay, the thermostat has been removed after checking the head gasket. The has been running without the thermostat and has been losing coolant before this and now.

Okay, I am wondering if you got a defective new radiator cap. If not check the sealing flange where the cap needs to seal against, I have seen dents or cracks in the flange which case the cap not to seal, also when the engine is warm squeeze the upper hose to see if there is pressure?

Okay, thank you. I'll get back to you after try all that.