It's probably not a stray...
This situation is all too familiar as something similar happened at my workplace about two years ago. The cat was being friendly, people started putting food out, the next thing the cat is a permanent fixture, waiting outside every morning... it turns out that it has an owner... the owner requests that people stop feeding it, but that doesn't seem to deter those putting food out...
Then came the fleas... followed quickly by the "STOP FEEDING THE CAT!" memo and after a week it vanished, never to be seen again.
The collar may have an ID with a phone number? If it doesn't then securely tie/tape a message around the cat's collar for the owner telling them what the cat is up to and see what comes of it...
Cats are quite wily creatures, not at all loyal like dogs - they will go to whoever feeds them and will often not think twice about abandoning their owner if someone else has a warmer fire and better/more food laid on...
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