If he belonged to a named group with political goals and motivations, I don’t think it would be unfair to call the prisons run by that group. I don’t think your comparison is as much of a “gotcha” as you think it is.
He is the leader of a party that is literally called Jewish Power (“Otzma Yehudit”) so yes, he runs a named group with political goals and motivations.
We don’t call Israeli prisons “Jewish-Power-run” just because they are currently run by Ben-Gvir. Maybe we should though.
If he belonged to a named group with political goals and motivations, I don’t think it would be unfair to call the prisons run by that group. I don’t think your comparison is as much of a “gotcha” as you think it is.
He is the leader of a party that is literally called Jewish Power (“Otzma Yehudit”) so yes, he runs a named group with political goals and motivations.