The official rule:
A fumble may be advanced by any player on either team regardless of whether recovered before or after ball hits the ground.
On a play from scrimmage, if an offensive player fumbles anywhere on the field during fourth down, only the fumbling player is permitted to recover and/or advance the ball. If any player fumbles after the two-minute warning in a half, only the fumbling player is permitted to recover and/or advance the ball. If recovered by any other offensive player, the ball is dead at the spot of the fumble unless it is recovered behind the spot of the fumble. In that case, the ball is dead at the spot of recovery. Any defensive player may recover and/or advance any fumble at any time.