No, events that come from external calendar integrations, such as Arbiter Sports, and are synced into iiQ, are not written back to other calendar integrations. Only events created directly within iiQ are eligible for outbound write-back synchronization.
For example, if an event is created in Arbiter Sports and then imported into iiQ, it will not automatically sync back to Google Calendar through the iiQ Google Calendar write-back integration.