Again, there are many ways to approach this question. Cockroaches who live outdoors are most active during spring and summer (specifically early to mid-summer).
Typically, roach infestations happen during these seasons because they love the subtle dampness of spring and since they cannot tolerate extreme heat, they would take refuge in houses and buildings.
If unabated and if spaces prove to have a stable food source, they would stay for a full-blown infestation. During autumn and winter, roaches tend to hibernate and will not be around so much. During this time, they would just be inside their habitats or beneath your furniture.