cleanupnetwork ok, first of all, try to use the
aam_edit_permalink capability. It was renamed in AAM 6.
aam_edit_permalink capability is just a second authorization layer on top of WordPress core. If your user cannot edit permalink, it is most likely he is missing some additional WordPress core capabilities. Try to add more content related capabilities to your user or role and this way you'll be able to determine which one is required.
Keep me posted.