My garage keeps charging me for work without fixing the ongoing problem - can they do this?

My car went to the garage for a service and because it was overheating. They serviced the car and said it was the radiator and a burst hose (repaired for £579). I got my car from the garage the next day, but the same problems kept occurring. I took it back and they said the radiator was faulty. The car was then ready to go again, so I picked it up car. I drove twice before the same thing happened again. The garage then changed the thermostat, but the same problem remains. They now say it's the head gasket and are going to charge me for repairs. Should they do this work for free as the original issue was never fixed?
Initially they fixed the reason why the car was overheating, which was a failed radiator/radiator hose losing coolant. However, if the cylinder head gasket is damaged, that is a consequence of the overheating that is another issue not fixed by replacing the radiator. The garage is doing the best it can, in the right order.
Answered by Honest John on

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