The speed of sound is measured in relation to the medium an object (or sounds waves) are moving.
The actual speed is dependant on the tempreture and some other factors of the medium.
Let's take for example the the information here :
From the EngineeringToolbox
We can see that in dry air, 20 celsius, the speed of sound is about 340 m/s.
In adtition, if we measure the aircraft's velocity with relation to the ground, we have to take in account the velocity of the wind.
In the example noted in the question, if we have lets say an aircraft travelling at a ground speed of 380 m/s, but it has a tailwind of 80 m/s, it means that by the ground track- it was supersonic if it would have traveled in relation to the ground. Meaning it does travel faster than the the speed of sound, if the sounds moves with no tailwind. But- in relation to the actual air (enclosed system), it travels in a velocity of 300 m/s, which is about 0.88 mach.
And that means- that a sound wave, in the same air mass- launched from the same origin with a base speed of 80 m/s (the tailwind)- would travel faster than the aircraft.