Well the Magento2 admin forms for websites, stores and store groups sadly don't use the eav model or ui components, so you don't have that much of a choice if you want to extend the form and/or the data which is saved/loaded using the form.
The form for the store groups is located in:
Due to nearly all functions beeing protected in the call tree, maybe the best solution, to add additional form fields to the store group form, is using a preference for the above class.
Following the call tree of
_prepareStoreFieldset function, you will see an additional function
_initFormValues, called after this one. This is a good place to load your additional attribute values, if this is even necessary.
If you than take a look into the saving process of the store groups located in
(yes it uses the same controller for saving of websites, stores and store views)
you see on line 93
So if you call your new store group input field, according to its name in the model, this should already handle the saving process of your new store group value.
btw. the store group model extends the AbstractExtensibleModel of Magento, means you also should be able to extend the store group's attributes by using extension_attributes.xml. But extension attributes also do NOT handle the saving and loading process for you. https://devdocs.magento.com/guides/v2.3/extension-dev-guide/extension_attributes/adding-attributes.html
(this answer is for magento CE 2.3.1, on other versions line numbers and so may be different)