The iron rod has high permeability. So if we insert the iron rod in the solenoid along its axis, the magnetic flux inside the solenoid will increases. The self-inductance of the coil is proportional to the permeability of the material inside the solenoid. Because of an increase in the permeability inside the coil, the self-inductance will also increase. Thus, a constant current I will be maintained in the solenoid. Then

1. magnetic field at the centre

2. Magnetic flux linked with the solenoid

3. self-inductance of the solenoid is increased if an iron rod is inserted in the solenoid along its axis.

Thus, option A, B and C is the correct option.