Okay, So I've had a chat with the Core member involved, I can't speak for the BHG guys as they aren't in the faction.
They logged to chase a BC, then saw a disproportionate number of hostiles log so they decided to fly directly up to avoid the gank.
The Core| BC then miss fired killing two of the BHGS ships. He then decided that they had flown too far away from any station or Jumpgate to fly back and choose to suicide. I am under the impression that none of your group actually saw/came into interaction with any of the ships listed above. I'm glad it wasn't suiciding during an encounter but it's still isn't good enough to reach the standards we want for Core|.
I've explained to the Core| member what they didn't do right here,
They understand the mistake they made and have given me their word not to fly off plain to avoid a spanking in the future, (doesn't matter if its 3 or 30) after all its just a game and dying doesn't hurt anyone, I'm confident that this specific player won't make this mistake again but the HC will be keeping an eye on them for a short while just to make sure they have understood the complaint and act on the feedback we have given them.