In a system, a pinion shift unit starts shift of a pinion to a ring gear for engagement therebetween during an internal combustion engine coasting in a forward direction after an automatic stop of the internal combustion engine. An engagement determining unit determines whether the pinion and the ring gear have any one of first and second positional relationships therebetween. The first positional relationship represents that the pinion is at least partly engaged with the ring gear. The second positional relationship represents that the pinion is in abutment with the ring gear. When an engine restart condition is met before it is determined that the pinion and the ring gear have any one of first and second positional relationships therebetween after the start of the shift of the pinion to the ring gear, a rotation adjusting unit adjusts a start timing of rotation of the pinion.