Can I put a 2006 cobalt ss supercharger on a stock 2006 cobalt engine.

Not without a bunch of mods. The supercarged engine is the2.0 LSJ. The non supercharged engines are 2.2. You will certainlly need the ECU and the fuel pump and probably the wiring harness and an intercooler to cool the intake air charge. The transmission in your car is likely a lighter duty version that won't like the extra horsepower without some upgrades. And if yours isn't an SS then it won't have the sport suspension which is nice when you've got all that extra horsepower. If its a project car you want I would suggest buying a superchared SS and building on that. Then you'll have something. Way cheaper than trying to make your own plus you can drive it right away instead of two years from now when you sort out all the electronic problems you'll likely have. There are still quite a few of these cars around and they are generally pretty cheap as the technology is getting a bit old and most people are obsessed with turbochargers.