How to mitigate change in amplitude when transmitting signal from coil?
Two coils that are magnetically coupled makes a transformer and, when you design that transformer correctly, there will be very little amplitude change across a wide range of frequencies.

So, the first thing you have to do when designing the transformer is to ensure that the primary impedance isn't so low that it messes with the amplitude of the signal source. Given that your signal source is a function generator with 50 ohm or 600 ohm output impedance, you are going to hit problems at low frequencies without winding hundreds of turns on the primary.

In your previous question you said this: -


  I am trying to transmit wirelessly a sine wave signal of frequency
  from 50 to 200kHz


So, is 50 Hz your low frequency or is it 50 kHz? That's my first question to you (the OP). Engineering requires absolute clarity because trying to drive 50 Hz across an air-cored transformer from a signal generator will require literally coils made from tens of thousands of turns. Trying to drive 50 kHz might mean a few hundred turns of 10 cm or more diameter - remember we are just talking about the primary coil here and, we are trying to ensure that your signal generator's output voltage isn't significantly lowered by the primary inductance being too low.

If your function generator has an output impedance of 50 ohm then the loading from the primary coil should have a magnitude greater than 250 ohm (just a rule of thumb). To get 250 ohm reactance at 50 Hz implies a primary inductance of nearly 800 mH and this requires many, many, many thousands of turns.

At 50 kHz the inductance that gives 250 ohm reactance is 800 uH in comparison and requires many fewer turns.

The trouble is that even with a few hundred turns (air core) it will become a resonant circuit at some point and you will get some disturbing and hard to counteract effects so, my advice is this: -

Take the output from your function generator and feed it into a moderate power amplifier that can sustain an output even when faced with a load impedance as low as (say) 5 ohms. This means that the primary inductance at 50 kHz (previously calculated at 800 uH) can be 50 times lower at 16 uH.

Of course, if you are insistent on operating at 50 Hz the former inductance of 800 mH becomes 16 mH and still requires hundreds of turns.

You can mitigate against some things by using a filter in the receive coil circuit that counters the loss of amplitude due to the primary loading but, give yourself a fighting chance and use a moderate power amplifier.