Sounds like an exam question to me...
As you climb through an inversion, the temperature increases
So pressure has dropped slightly, and temperature has increased. Overall, the density has gone down. TAS = RAS (I call it CAS myself) divided by root sigma. Sigma is relative density, so in this case it's increased.
This means that at constant TAS, CAS / RAS will go-down. In practice however you fly at CAS, which is directly related to IAS, so if you keep the same number on the dial, you'll go faster.
Local speed of sound is proportional to the square root of absolute temperature, so when it gets warmer, the local speed of sound will go up, and at constant TAS, the Mach number will consequently go down. Not by very much mind you.
Descending through an isothermal layer at constant TAS (again you wouldn't, you'd fly at constant CAS) Mach number will remain constant because it's a function of temperature, not density. Because density is increasing, CAS / RAS will reduce at constant TAS.
Which exam was that out of? And who thinks you fly at a TAS rather than a CAS / RAS / IAS ?
G