Contacts sync via CardDav misses some
As far as I know this is a default behaviour of Google Mail.
On Android there is a fairly hidden way to select more groups but not on iOS or other normal cardDAV clients.

You may find fruux an alternative that works fine for me in this case.
If there is a single addressbook at fruux with a default name like “contacts”, iOS will attempt to interpret vcf-Groups as contact groups. E. g. eM Client will create contact groups for each new assigned category simultanously.