Read last paragraph for more context and ancillary information. It may not be necessary though. (Trying to respect your time)
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Tasks come into frontdesks (GrpA) for the maintenance department(GrpB). We have Office 365 Enterprise1. I want to create a shared resource I call "The Bucket". All members of both groups can add/edit tasks but GrpB can't delete anything.
I also want a unified mailbox called bucket@mydomain.com where we can funnel all inbound emailed tasks then convert those emails to tasks.
End result, everyone has one list of everything to be done and the status of the tasks, see the bucket emails. I do not want maintenance to be able to delete emails or tasks.
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I manage beach houses with the help of an awesome frontdesk squad and maintenance department that is lacking in task management skills. I also have many many hours in research and trying to have the exact end result described below. I've tried several ways but always seem to be missing one piece. I'm hoping someone here has global knowledge of shared mailboxes, public folders and exchange in general so the can tell me which is the most logical way to set this up. I love tinkering and figuring things out on my own but I can't put any more time in this. I'm getting behind in my job-job. I know just enough about this stuff to really sound dumb.
I would consider something like granting the Author level permissions on the tasks folder for GrpB, or simply granting a custom set of permissions. You can review the available options here: http://technet.microsoft.com/en-us/library/ff522363(v=exchg.150).aspx
It's not necessary to adjust them with PowerShell, you can just right-click on a folder in Outlook or OWA and do so.