Client groups are not considered legal entities in Karbon (they are a collection of entities) and do not have email addresses associated with them. Because of this, there is no email to send invoices where the client group is the 'Billling Entity'.
For work or time that is assigned to the client group, you can select a public person or organization contact to bill these items to. To do this, go to Details > Billing on the client group and select a 'Billling Entity' contact from the search field.