Transmission drops to second gear and stay there, not shifting at all

The first thing that should have been done was to read and record the diagnostic fault codes. That will give you an idea of why the system has been going into 'limp mode". Limp mode is where it stays in second gear until you cycle the ignition switch off and back on. Most commonly this is caused by a defective sensor or solenoid, or a problem with the associated wiring. Those defects are often detected right away, and the system goes to limp mode before the car even starts moving.

There can also be excessive wear to the clutch plates that result in slipping during shifts. That slippage is detected while driving, so it doesn't go into limp mode until the car is moving.