Fine structure in the F Rydberg states of indium has been studied by optical-microwave doubleresonance spectroscopy. Microwave transitions between the neighboring levels of the F5/2 and F7/2 Rydberg series have been measured for the range of n =35, 36, and 39W1. The experimental results show that the fine structure of the F states is inverted. The observed inversion in the fine structure is found to be closely correlated with a long-range interaction between the inner core and the valence electron of the atom.