I had a similar problem w/ my '02. It would not stay in 4x4 in reverse. At first I thought it might be internal in the front diff. After removing the front fenders, I decided to try something easy first. I ended up adjusting the engagement lever on the front diff to the rear a couple of notches and then reinstalled and readjusted the shift cable to compensate for the difference. Hope this helps!
The 2001 Suzuki Grand-Vitara has rear wheel drive.
The 2001 Suzuki Grand-Vitara has a 8 ft. 1.6 in. (97.6 in.) wheel base.
The 2001 BMW M5 has rear wheel drive.
The 2001 Acura TL has front wheel drive.
The 2001 Audi A6 has front wheel drive.
The 2001 Ford Expedition has rear wheel drive.
The 2001 Toyota Camry has front wheel drive.
The 2001 GMC Yukon has rear wheel drive.
The 2001 Subaru Forester has all wheel drive.
The 2001 Audi A4 has front wheel drive.
The 2001 BMW X5 has all wheel drive.
The 2001 Ford Mustang has rear wheel drive.