The flow of a curve is said to be inextensible if the arc length in the first case and the intrinsic curvature in the second case are preserved. In this work, we investigated the inextensible flow of a curve on $S^2$ according to a modified orthogonal Saban frame. Initially, we gave the definition of the modified Saban frame and then established the relations between the Frenet and the modified orthogonal Saban frames. Later, we determined the inextensible curve flow and geodesic curvature of a curve on the unit sphere using the modified orthogonal Saban frame. Also, we gave some theorems and results for special cases of the evolution of a curve on a sphere. Finally, we gave examples and their graphs for the inextensible flow equation of curvatures.