In the fictional book I'm writing there are the institutions, Focs Schools and Focs Net. These institutions greatly distort history, science findings, etc. Can I get sued by Fox News for the fictional name, when pronounced, sounding like the real life counterpart?
Personally, if I were you, I would ask a lawyer versed in defamation and civic law more so than a fiction writer (or do some legal research on court cases about that problem, which can be quite dense and dry). However, legality aside, I would maybe recommend making the tie between your fictional news outlet/institutions and Fox News more vague or nuanced. As I read it now, it's a little obtuse. I don't have to do much thinking or reading to see who the target of your writing is. You're obviously reflecting the badness of Fox News, if not out right satirizing or parodying them, and you don't need to have a name that is phonetically similar to the aforementioned news outlet to do that well. If anything, change the names of your institutions to not even remotely sound like Fox News (emphasis on the Fox part). After all, it's not the name that's the problem, but the conduct. If you adequately satirize or parody, or just reflect, the conduct of Fox News and the consequences of said conduct, then your audience, assuming they're acquainted with Fox News, should make the connection without having a thinly veiled name telling them the connection outright.
Hmm, I might point out that Fox News is a major juggernaut with a competent legal arm.
When I trained as an EMT, one point that the instructor drummed into our heads was, "50% of all lawyers graduate in the bottom half of their class." He said (whether this is exaggeration I don't know) that someone can sue you for having the wrong color shoelaces. The case will get thrown out of court, but it's still a hassle for you.