The groundspeed can be more, less or the same as the airspeed.
Go back to boats. I'm driving around in my boat on a big river with a 5 mph current. My boat goes 30 mph. If I'm going with the current, my boat is going 35mph relative to the river bottom and the shoreline. If I turn around and go the other way, my boat is going 25mph relative to the river bottom and the shoreline. If I go across the current, it's a wash and my boat speed and speed over the river bottom is the same, as if there was no current.
My boat's water speed is 30 all this time. My boat's "river bottom speed" is water speed plus or minus the effects of the current. As far as the boat itself goes, it doesn't know or care what direction its going relative to the river bottom, it's just concerned with the speed relative to the water surface, which is moving relative to the bottom.
An airplane is flying along a "river" of air and it has its airspeed, the speed through the air mass, and it's speed relative to the surface the air mass is flowing over, the "river bottom", or the ground in our case.
The pilot's main concern with groundspeed is the same as the boat captain's concern with boat speed added or subtracted from the current (say when navigating in the ocean). It affects how long it will take to get from A to B, and how much fuel will be required, and related things.
It doesn't have much effect on the flying of the plane itself except in an extreme case like a windshear encounter, where the inertia of the airplane itself can be a factor. The case you describe could be what is called a "reduced performance shear", which is a very sudden and extreme tailwind, which makes airspeed drop while groundspeed stays the same or increases fairly slowly. An airliner on approach that ends up with a downburst behind it, and is caught in the high velocity outflow of the downburst from behind, would see that happen due to the inertia of the aircraft until it's able to recover it's original speed through the air mass. While it's doing this it would have to pitch up and use some of its inertial energy to keep from descending below the glide path.
Even there though, you aren't really concerned with the groundspeed itself, except to the extent that the groundspeed affects how quickly go from A to B on the approach. You are totally concerned with recovering the lost airspeed that the airplane needs to fly.