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What page does this menu point to?

In general, creating a new menu item and pointing it at an admin page doesn’t automatically give users permission to access that page. A custom menu is just like a normal link in this respect. You can use the capability settings in AME to control who can see the menu, but that doesn’t affect who can see the page that the menu links to.

The solution is to give the author role the capability (or capabilities) that is required to access the page. The free version of AME can’t do that. The Pro version can mostly do that automatically for pages that are part of the normal admin menu. For other pages on the same site, you might need to use a role management plugin like User Role Editor. For pages on other sites – i.e. external links – you’ll need a custom solution.