With one of those funny “only let me in” collars (if you can get one on said Mog of course), then if you find it comes back without said collar you’ll know it’s got another family closeby..
One of those funny collars that the cat loses and then gets stuck outside in the cold winter!
You not got a family or someone that could feed it? Can you puit a cat flap into a shed door if you don’t want to do it into the hosue? We took our to the cattery once but it cost quite a bit and he hated us when we got him back. Now leave them to the run of the house with the cat flap and we have someone come and feed them morning and night