Who is responsible for an external QA when a service provider has been contracted to provide total outsourcing of the internal audit activity?

In all cases, the organization maintains the responsibility for having an external QA in accordance with The IIA's Standards. If the organization has a CAE (partial outsourcing), it is clearly the CAE's responsibility to initiate the process and discussion with the audit committee. If total outsourcing exists, the person who negotiates the outsourcing of the internal audit services (e.g.,  CFO, Corporate Controller) would be responsible for initiating the external QA. The service providers' specific work on the assignment would be reviewed as part of the external QA and not the entire firm's policies and procedures (except relevant section of the Policies & Procedures of the service provider as applied in the organization). Service providers must advise and brief their clients on the requirements of the Standards.