In a system installed in a vehicle, an automatic stop of an engine is carried out when an engine automatic stop condition is met during the vehicle being decelerated. The automatically stopped engine is restarted when an engine restart condition is met before the vehicle is stopped. A vehicle speed obtaining unit obtains a vehicle speed, and a threshold determining unit determines a threshold based on a predetermined-timing value of the vehicle speed. The predetermined-timing value is obtained at a predetermined timing when or after the engine restart condition is met. The threshold is used to determine whether to enable a next automatic stop of the engine. A stop enabling unit enables the next automatic stop of the engine when the speed of the vehicle obtained after the engine restart condition is met is equal to or higher than the threshold.