Is your car petrol or diesel ?.
Is there any traces of oily sludge in the coolant reservoir or any mayonnaise looking sludge on underside of the oil filler cap. The investigation will most likely involve a compression test and or a sniff test.
I would most likely drain the coolant and flush the system, refill with new coolant and check it again after a few days to see if there is any traces of oil in the new coolant, if any oil does appear in the new coolant then your head gasket is kaput.