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 Hyundai Elantra has front wheel drive.
The 2001 Acura NSX has rear wheel drive.
The 2001 Chevrolet Camaro has rear wheel drive.
The 2001 Hyundai Sonata has front wheel drive.
The 2001 Volkswagen GTI has front wheel drive.
The 2001 Toyota Prius has front wheel drive.
The 2001 Volkswagen Passat has front wheel drive.
The 2001 Volkswagen Jetta has front wheel drive.
The 2001 Nissan Frontier has rear wheel drive.
The 2001 BMW X5 has all wheel drive.