The true airspeed (TAS) can be calculated from the indicated airspeed (IAS), which is derived from the pitot tubes and static ports, as follows:
$$ \mathrm{TAS} = \mathrm{IAS} \sqrt{\frac{\rho_0}{\rho(a)}} , $$
where $ \rho_0 $ is the air density at sea level and $ \rho(a) $ the air density at altitude $ a $, which depends on pressure $ P $ and temperature $ T $:
$$ \rho(a) = \frac{M \cdot P(a)}{R \cdot T(a)} , $$
where $ M $ is the molar mass of air and $ R $ is the universal gas constant.
Using the international standard atmosphere for $ P(a) $ and $ T(a) $, one can plot the TAS as a function of altitude:

The ground speed (GS) is then given by the vector addition of the TAS and the wind speed:
$$ \mathrm{GS} = \mathrm{TAS} + v_\mathrm{wind} \cos(\alpha) , $$
where $ \alpha $ is the angle between the wind direction and the track of the aircraft ($ \alpha = 0^\circ $ for tailwind, $ \alpha = 180^\circ $ for headwind). This step is independent of pressure or temperature and as such independent of altitude. This means for a given TAS and headwind component, the ground speed is the same at all altitudes (for your example: 100 mph GS with 20 mph headwind implies 120 mph TAS at all altitudes).