What does and EGI rellay do???

[sr20tom]

After the summer my wiring got really weard. When you take the key out, the lights and the motor keeps runnig for another second or two. After the car shuts off and the cluster lights go off another second later, my EGI rellay starts to wind. If i take the EGI rellay out, the car runs and drives fine and shuts off right after i turn the key off. But there is a change in the air fuel. At idle without it, the car runs in the 11.5s and if i put a rellay that i borrowed from a friends car, it idles in the high 13s and low 14s. What is the function of that rellay?