Based on the '06 GT I used to have, there are a few things you'd want to check:
- Clutch-type limited slip differential was standard on the GT in '05 and '06. There was a TSB out for it because of chatter during tight low-speed turns (e.g. parking lot maneuvering) because Ford had used a sub-optimal differential lube initially and the TSB involves replacing the clutch packs in earlier models. You'll want to make sure that this was taken care of.
TSB 07-22-08.
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Fuel tank filling issue. On the '05 and '06 cars the fuel tank would sometimes pressurize while filling and cause the fuel pump to hit its auto shut off before the tank was even 3/4 full. This was annoying, but could often be prevented by fueling with the car facing downhill or by angling the fuel filler nozzle differently.
- Fuel pump cavitation issue. After an extended cruise, the car will hesitate or shudder when you slow down and then try to accelerate again. The issue is caused by cavitation at the fuel pump (air bubble build up in the pump and when you slow down the engine experiences a momentary fuel pressure drop which causes a stumble). Ford has a TSB out on this and it's
TSB 06-9-9.
My '06 experienced all of these issues, with the last one being the most annoying because the dealership refused to repair the issue stating that they could not replicate it (because they would not take the car for the multi-hour highway cruise necessary to replicate the issue).
All that said, I really, really enjoyed that car.
ZV