It appears that the laptop is configured to access the Internet from the school's network, which is actually on par with how many networks set up their laptops.
I wouldn't change the domain if I were you
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The Cisco VPN client has its own firewall on it which blocks all traffic going from you to outside the school network if the VPN is turned on.
The workgroup thing is only for file sharing.
You want to open a command prompt when at home and type ipconfig /all and see what it comes up with
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Mitch