Lincolnshire Police identified the people in a series of CCTV images after criminal damage was reported at a shop in south Lincoln.
The incident happened on Monday, May 27, between 1.54am and 2.07am at the Lincolnshire Co-op store on Lincoln Road in North Hykeham.
Three young men and one woman were seen at the scene. One man stamped on a sign near the shop entrance, then threw stones at the CCTV camera.
Another man then punched the CCTV camera, causing damage to the equipment.
Later on Friday two people have handed themselves in and Lincolnshire Police said the incident has been dealt with via restorative justice.