This SOP covers how a QA group member can apply to become a sponsor, and how existing group sponsors and administrators should handle such applications.
To apply to become a group sponsor, file an issue in the fedora-qa issue tracker. If you can, apply the sponsor tag to the issue. In the description, explain that you would like to help out with reviewing group membership requests, and affirm that you have read the SOP for handling membership requests. Please understand that this is the only reason to become a sponsor - being a sponsor does not grant any other special powers or privileges.
Existing group sponsors should keep an eye out for applications to become a sponsor: this link will search the tracker for open issues with the sponsor tag. In general, applications from existing active group members should be accepted. Applications should only be rejected if they come from applicants who don't appear to be active group members at all, or if there is some specific reason to be concerned that the applicant will not review new membership requests correctly.
If you believe the application should be rejected, post a comment and explain why. Any group member can raise an objection, and the existing sponsors and administrators will review them. You can also post a comment explicitly endorsing the application, but it is not really necessary to do so.
Once an application has been open for a week, if there are no objections, any existing administrator can (and should) accept the application, by logging into FAS, setting the applicant's role to sponsor, and then closing the ticket with a comment that this has been done. If any objections have been raised, existing sponsors and administrators can use the ticket to discuss and ultimately come to a consensus or a vote on whether to accept or reject the application.