There are 4 or 6 inch hole drill bits but they can only go about 2 inches deep before they bottom out. then you would be stuck with figuring out how to get that 2 inches of material out so the drill can go another 2 inches but I feel like there has to be a tool for drilling deeper.
https://www.amazon.com/IRWIN-373400BX-4-Inch-Bi-Metal-Hole/dp/B0002UKVA2 This is what im talking about. They might go deep enough that you would only need to get the material out once before you could totally clear it.