BTW, I'm curious to hear more about the mechanics of your scenario. The AGI hacks itself onto every (Internet-connected) computer in the world. Then what?

Then the AI does precisely nothing other than hide its presence and do the following:

Send one email to a certain nano-something research scientist whom the AI has identified as "easy to bribe into building stuff he doesn't know about in exchange for money". The AI hacks some money (or maybe even earns it "legitimately"), sends it to the scientist, then tells the scientist to follow some specific set of instructions for building a specific nanorobot.

The scientist builds the nanorobot. The nanorobot proceeds to slowly and invisibly multiply until it has reached 100% penetration to every single human-inhabited place on Earth. Then it synchronously begins a grey goo event where every human is turned into piles of carbon and miscellaneous waste, and every other thinghy required for humans (or other animals) to survive on earth is transformed into more appropriate raw materials for the AI to use next.

And I'm only scratching the surface of a limited sample of some of the most obvious ways an AI could cause an extinction event from the comfort of only a few university networks, let alone every single computer connected to the Internet.