I have a older sears gas furnace and the piliot light goes out when the furnace turns off. What could be the matter.

The pilot orifice could be partially restricted causing the pilot outage when the main gas kicks off. In this case the main gas flames when shutting off kind of snuff the pilot out. Besides checking for a clean orifice also check the pilot gas pressure and makew sure there is a good steady pilot flame.

Next is to check the burner gas pressure setting and the air mixture shutters. Adjust for proper CO/temp reading.

Also on a older furnace it is a good idea to have the heat exchanger checked while the service person is there doing the rest of the work.