How can I prevent constant reboots?

My Windows7 installation has been becoming less stable for some time. Then very recently it began to experience unexplained and very annoying reboots which became increasingly frequent. So I reinstalled it from CD using the route.After some initial problems - it kept seeing programs it wanted out of the way, even after I had uninstalled them - it ran fine, completed, and validated - but the reboots didn't stop, they got worse!!
The odd thing is that the nature of these events has changed. At first they had nothing particularly to do with going online, they just happened randomly whatever I was doing. Then in the next stage if I ran AVG's One-click maintenance as soon as I booted up, it would be stable for a few hours, but then it got to the point where it would crash before One-click could finish! One-click won't run in safe mode!
So then I could operate only in Safe Mode with Networking and I could go online, but now I can't and even if a program tries to go online to check for updates, Windows7 immediately reboots. Even if I am in safe mode without networking, if a program just tries to go online, even though that's impossible, the try causes a reboot.So I cannot use Dr. Hardware or Device doctor etc.
I know that there is a debugger which needs dot NET framework v4 in order to run and I managed to download both initial files, but neither will install without access to the internet and if they try that then we get a reboot.
I have a 64-bit system with 64x2Dual Core Processor 4200+ 2.13GHz and 3.00GB memory.
The problem is not overheating of the CPU; the PC will run all night in safe mode if untouched or if in use, though if left and then woken up it might throw a fit. On the other hand in full use mode it won't last five minutes now. Nor is it a problem with the internet connection. The smaller PC, on which I'm writing this, is connected to the same BT boxes by cable and there are two others connected wirelessly. None of them are experiencing this problem.
Can someone please tell me how to get out of this dilemma.

In addition AVG has not been ever recommended here, since it bloated up and became buggy before Win7 was even in beta. I'd uninstall it and use Microsoft Security Essentials.

As with all bloatware AV's it requires a special Uninstaller (removal tools) for common antivirus software - ESET Knowledgebase

Compare the install you have to the perfect install in Clean Reinstall - Factory OEM Windows 7 (same for retail) to see if you could improve it by reinstall.