Inductance of air core coil decreasing due to ferromagnetic object put in it?
When I put an iron core or any other ferromagnetic object into the
originally air core coil L2, the output frequency will increase and
the voltage will decrease.

If the inserted iron or ferromagnetic object is not laminated or made from ferrite (which doesn't conduct electricity) then it will act like a shorted turn due to eddy currents and the net inductance of coil L2 will fall. This will cause frequency to rise.
This is why power transformers use laminated steel in their core construction and it's why ferrite rod antennas are made from ferrite material.
It's also why metal detectors can find gold and silver (well, any conducting material) - magnetic induction creates small voltages in the &quot;target&quot; metal and, these small voltages circulate eddy currents. The eddy currents create a change to the magnetic field and this is detectable.