In the legacy admin interface, when creating/editing a content type, it's possible to put each field definition in a field group ("Category" in legacy vocabulary). Those groups are defined in site.ini/[ClassAttributeSettings] :

[ClassAttributeSettings]
# List of attribute categories, by default used for grouping
# attributes in edit interface.
# Key is identifier and value is used in UI as name
# Note: Limit on identifier is 25 charters!
CategoryList[]
CategoryList[content]=Content
CategoryList[meta]=Meta

# Default Category for attributes without or with invalid category
# Note: Must be defined in CategoryList!
DefaultCategory=content

In the REST API (and I guess it's the same at the Public API level), there are two issues:

1. if in the admin interface, I let the "Default" category, in the REST API the fieldGroup property is empty (and there's no way to retrieve the default category)
2. if I explicitly set the category in the admin interface, the fieldGroup is filled with the category identifier but there's no way to retrieve the category name