Missing Global Everyone Group

convert the drive to ntfs:
convert c: /fs:ntfs

Article ID: Q149091 Power Users Can Modify Common Groups
SYMPTOMS
Only administrators should be able to modify common groups in Windows NT Server and Workstation. However, members of the Power Users group are able to add, delete, and modify icons in common groups. Power Users can also create new common groups and delete existing common groups.

This problem does not occur on NT 4.0 Workstation or Server when installed on an NTFS partition. Default permissions on the %systemroot%\profiles \all users directory which defines common groups and icons grants full control to the Administrator and System acconts with read-only access to the everyone group. NT 4.0 Workstation and Server installed on a fat partition are subject to the same limitation as 3.5X.

WORKAROUND
To resolve this problem, upgrade to Windows NT Workstation and Server version 4.0 with on an NTFS partition..

You can work around this problem by not using the Power Users group.

STATUS
Microsoft has confirmed this to be a problem in Windows NT versions 3.50 and 3.51. This problem was corrected in Windows NT Workstation or Server version 4.0 when installed on NTFS partitions.

LOL...The group Everyone is not a built-in domain group, therefore it is not a global nor local group you can modify.
The Everyone designation is just that....everyone or anyone who has access to your LAN. This group is not an authenticated group within the domain. It is not unique to any domain since it is a member of all domains.
Point is..there is nothing to modify because it doesn't exist in your domain.

Try this:

You'll see every one group when you assign permissions on files / directory not in user manager.

If still not exists then you can use Authenticated Users instead of every one to assign permissions.