The FBI only has branches in 56 cities in the United States. Where the FBI has no branches, if they handle cases, they can only rely on the help of local police. Let me give you a recent example, which happened this month: On February 2, 20021,FBI agents went to Hideichi, Florida to arrest a suspect, but they were resisted by armed men. Two FBI agents were killed and three were injured. Because the FBI didn't bring a swat team, the swat team of the local police (county sheriff's office) was the first to come to the scene for support: in daily work, some small local police stations don't have enough technical resources to conduct criminal investigations, so sometimes they will ask the FBI for help, but sometimes they will ask the state police for help.
Police in big cities generally don't ask the FBI for criminal investigation techniques, but the FBI often takes the lead in setting up a task force to crack down on some crimes, and its members will include state, county and city police. At this time, the state, county and city police can call themselves the FBI, because they are temporarily led by the FBI.
The jacket worn by the man on the right is the most telling: it says FBITaskForce, which means this is an FBI task force, but it says State Police below, which means he usually works in the state police department. The jacket on the left only says that the FBI is usually an FBI agent.