Suppose you had 2 equal point masses, one at r and one at 2 r. The moment of inertia is m rsq + 4 m rsq or 5 m rsq. If I place both masses at the center of mass at 3/2 r, then the moment of inertia is 2 m x 9/4 rsq or 4.5 m rsq. I think this is where the confusion lies?
proposed problem
This might help clear up the question. Suppose a force F is applied perpendicular to the rod at a distance of 1/2 the location of the center of mass from the center. What is the angular acceleration?