We are using Google Apps and want to remove one of the email addresses from the "everyone" list. Imagine a helpdesk ticket being created every time someone sends an email to everyone.
What options do I have in accomplishing this?
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It's not possible to exclude accounts of the domain of the everyone list.
You can't edit the members within the All users within your-domain member. But just like any other group, you can edit the groups access settings or delete it.
Note: On 2018-07-17 the above quote continue being valid.
Instead, create a group to which you add the desired members by other means. There are several options for that.
Another note: Suspended users will not receive emails sent to a "everyone" list.
It should be possible to do this now, using Content Compliance. See this answer: https://webapps.stackexchange.com/a/122559