When adding "Survey Editor" OOB role to a user, the user is able to export survey results. 
When adding survey permissions to a custom role, and assigning the custom role to a user, the user is unable to export the survey results. The user receives error "Survey export failed. Try again"

Cause

"Survey result" permission is added to the "record type" section
 

Resolution

Add only the following permissions to the custom role: 
Agent portal>Administration>People>Roles>{choose custom role}
Record type:
Survey: mark "Create" / "View" / "Update" / "Delete"
Survey question: mark "Create" / "View" / "Update" / "Delete"
Resources:
Survey result: mark "Create" / "View" / "Update" / "Delete"

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