AEM 6.3 introduced Manage Publication for sites. As per what I have observed this is only available for admin and not content-authors group in site admin for touch UI
Is there anyone who has added permission for Manage Publication to any other group?
 Adobe Experience Manager Help | Understanding Publication Management with AEM Sites
Appreciate your help.
I got the solution for this. When you need this option for any group, make sure it has /etc/workflows/models read permission.
Thanks for the reminder, I should have updated
I'm having the same problem where I need to enable "Manage Publication" menu on our authors.
Following your solution shows the menu but it opens to a new problem.
Now, "Manage Publication" menu is showing even though users/groups have no replicate permission.
This problem address to smacdonald2008 and kautuksahni too.
Yes, it needs to have access to workflow models in order to request for Publication.
it needs to have access to workflow models in order to request for Publication
Let me have a look at this one and get back to you.