Can I group fields by their Field Group in Views?
Field group doesn't actually do anything with regards to data storage or grouping, it only provides fieldsets and formatters for data entry and grouping fields together on display.

It seems to me that you'd find either Field Collection or Tablefield more useful for this use case.

With TableField, you can just output the field in Views, as it's already a table. 

With regards to Field Collection, it's a little more work, but much more flexible. You should be able to add a relationship in Views to the the field collection entity that contains your fields, and then group by the entity ID of said field collection in views. 

You will also probably have to use something like Views Field View to embed your grouped Field Collection fields into your original view if the other fields aren't to be laid out like tables..