Ouch !!!
A sinkhole is a serious problem. If the sinkhole is large enough then it could destroy the value of the property. Not only would it not sell, but it poses a risk to anyone living in the home.
Hopefully this is a “smaller” sinkhole. The homeowner should contact a geophysical engineer or other experts that have knowledge of sinkholes. If the homeowner is lucky then contractors can fill the sinkhole with dirt and gravel.
Unfortunately, we probably cannot help this homeowner. If the home is resting on a pier and beam foundation we may be able to place some additional concrete pads under the affected area. But we would be very hesitant to drill holes in the vicinity of a sinkhole.
Sorry. If you own a home near a fault or a sinkhole then you have a major, major problem and may not be able to sell your home under any circumstances.