How do I make a Group-Only spawn?

You would check if they're in the group that the user is in using the 'IsInGroup' method of Player.

Some code would look like this:

local groupId = 0 -- Replace this with your group Id.

game.Players.PlayerAdded:connect(function(plr)
    if plr:IsInGroup(groupId) then
        -- Either teleport them to the 'spawn'
        -- Or change their TeamColor to a 'Group' Team's TeamColor.
    end
end)

EDIT: If you're teleporting them to the spawn, don't forget to wait for the character to load;

plr.CharacterAdded:wait() -- waits for the 'CharacterAdded' event to fire. This would signal that the player's character has loaded and there should be no errors when you teleport them.