Who is the owner in this case and does that account have permission to execute sp_send_dbmail?
For the trigger itself, It looks syntactically correct but I am not sure the logic is really what you are looking for.
IF (select count(*) from boardbatchwhere graphgroup = 'BAR') = 2
Do you only want to be notified when the count = 2 or do you want to be notified when any group gets update to 'BAR'? You should probably look at using the inserted table for this.
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