Ztruker has given you a competent means for sorting out some of the diagnosis of this problem.
In the doing of it, it is important to recognize that when booted into SAFE Mode /w Networking, your machine's basic security programs of AntiVirus and Firewall are turned-off and not actively protecting you.
Therefore it is risky to do much of anything related to the internet, browsing and email.
If you are going to run your machine for extended periods of time or use a variety of your available processes and applications while in SAFE Mode /w Networking.........,
It would be prudent to Disconnect the Machine physically from the Internet and disable any Wireless Internet connectivity.