When a Project is running as part of a programme, Quality Assurance is responsibility of?

It depends on the company. 

In case of projectized organization it will be the PM of the project but for other instances Program Manager will have more control and can assign either the Project Manager to use his resources for QA or have company resourced QA team work on it. 

The basis of this decision may be inspired by company culture as all QA must be done external to the project or financial that QA is the part of the project and must be done by the project team or just the availability of the QA resources

A Project Manager (PM) will have overall responsibility for the successful planning, execution, monitoring, control and closure of a project.They must have a combination of skills including an ability to ask penetrating questions, detect unstated assumptions and resolve conflicts, as well as more general management skills.

I think B-Option project coordinator is good answer.

c. Programme management external to project

i think it should be the project manager's responsibility because his duty is to deliver the right required outputs with the desirable quality  

I think it's the product delivery manager.