Why does the cooling fan keep coming on and off every three to five seconds?

The first thing we need to do is check the fuses #46, 45, and 42 in the under-hood fuse panel. If they are all good, I would run a CAN scan to see what comes up because it can be a bad PCM, instrument cluster or one of the two engine coolant sensors. You can get a CAN scanner from Amazon for about $30.00. here is a video to show you how:

https://youtu.be/u-4syLc-ifQ

Here is a scanner link:

https://amzn.to/3o41KcZ

and

https://www.2carpros.com/articles/how-to-check-a-car-fuse

You might want to try to unplug one coolant sensor one at a time to see if the behavior stops. Here are the locations and the wiring diagrams for the cooling fan in the images below. Check out the diagrams (below). Please let us know what happens.