How to add Active Directory user group as login in SQL Server

You can use T-SQL:

use master
GO
CREATE LOGIN [NT AUTHORITY\LOCALSERVICE] FROM WINDOWS WITH
DEFAULT_DATABASE=yourDbName
GO
CREATE LOGIN [NT AUTHORITY\NETWORKSERVICE] FROM WINDOWS WITH
DEFAULT_DATABASE=yourDbName

I use this as a part of restore from production server to testing machine:

USE master
GO
ALTER DATABASE yourDbName SET OFFLINE WITH ROLLBACK IMMEDIATE
RESTORE DATABASE yourDbName FROM DISK = 'd:\DropBox\backup\myDB.bak'
ALTER DATABASE yourDbName SET ONLINE
GO
CREATE LOGIN [NT AUTHORITY\LOCALSERVICE] FROM WINDOWS WITH
DEFAULT_DATABASE=yourDbName
GO
CREATE LOGIN [NT AUTHORITY\NETWORKSERVICE] FROM WINDOWS WITH
DEFAULT_DATABASE=yourDbName
GO

You will need to use localized name of services in case of German or French Windows, see How to create a SQL Server login for a service account on a non-English Windows?

In SQL Server Management Studio, go to Object Explorer > (your server) > Security > Logins and right-click New Login:

Then in the dialog box that pops up, pick the types of objects you want to see (Groups is disabled by default - check it!) and pick the location where you want to look for your objects (e.g. use Entire Directory) and then find your AD group.

You now have a regular SQL Server Login - just like when you create one for a single AD user. Give that new login the permissions on the databases it needs, and off you go!

Any member of that AD group can now login to SQL Server and use your database.