After that is complete, do I notify the auditors to do a check up or do I submit this information to them?
If it's 3rd party CB stuff it depends on what the CB requires, not the auditor. Where I do my work the clients are provided an internet "portal" to post the CA's and an automatic notification is sent to the auditor or whoever is the designated person to review "proposed" CA's to accept or decline. In the case of acceptance, the client receives an automatic notification as such and they proceed to implement and perform verification of effectiveness prior to the next scheduled audit when everything should be closed out (you still have to do your implementation, verification and closure, failure to do so can be ugly).
Your CB should be telling you this and instructions should be in the body of the report. Everyone does it different. Your auditor can't tell you how to correct anything, but he should tell you the expectations of what must be in place. Done it 1000+ times.