Can you have a single halo game with Xbox live, and system link

First of all, my question was a little bit complicated so I had trouble with the title. Feel free to edit.

I was reading some other posts and noticed most people said halo was one of very few games where you can have multiple people on multiple consoles.

This brought an interesting question to my mind: If you have an x-box 360 (call it console A) system-linked to another x-box 360 (console B) and console A is also using x-box live to host the game for another x-box (console C), would console B also need Xbox live to join the game? What if console B or C was hosting the match?

To make the question a tiny bit less complicated, I am limiting this question to the halo series, preferably halo 4.