The setup description in post #1 is vague in several regards. Most of all, which quantity does it consider as "inductance" , how is it measured?
Basically, by replacing air by ferromagnetic material in a magnetic circuit, you'll increase the inductance, at least for low frequencies. Core saturation might counteract. Also if you measure the coil AC impedance at a certain frequency, you don't see pure inductance, also eddy currents and other loss terms.
All in all, it can be calculated what happens in a well defined magnetic circuit, but it can be hardly predicted what happens in the partly unknown setup.