There is a way that you can have the system NOT send a message to someone in a group.
If you go to User Administration and expand a user. Click on a group they are a member of and select the action to modify user group. There is a check box to "Notify User". Make sure that is selected.
Also, on the user record, confirm they have External as their notification method. This could also prevent them from getting the notice.
Another thing, in your screenshot, when you put anyone in any of the 3 boxes (Group, Role or User), you need to select the "Notify Specified Recipients" box as well.
Apologies for the delay on this response.
Activating the option "Receive Group Messages" did the trick.