Nested teams within GitHub do not automatically inherit a parent team's context permissions.
If a parent team is selected as a security group for a context, child teams are not automatically granted access.
If a child team is not listed as a security group you will get an "Unauthorized" error when triggering a build in CircleCI's UI please see the screenshot below for an example.
In order to grant a GitHub child team access to context, you will need to add the team as a security group for the context.
For more information on contexts please refer to our official documentation: https://circleci.com/docs/2.0/contexts/