Alkonium

No, they wouldn't serve the Undying Sith Empire; they aren't Sith, in reality they dislike and even despise the Sith, much of what the Sith do are against the 'Nex Venator's' philosophies - even if they themselves aren't all too morally sound.

They, for the most part, are loyal to the Empire. Though, yes, if the Republic were ever weakened (and the Jedi along with it) they would exploit them, though secretly, they're a shadow organization for the Empire, not blatantly obvious saboteurs.

Their true and primary purpose is to ensure the Empires' survival by any means, and to ultimately protect it's current heir and ruler (lest that ruler be a usurper, in which case they would do everything in their power to get the usurper off the ruling seat, especially a Sith), seeing as they are based off of the late Emperor Palpatine's Shadow Guards (some of the founding members were Shadow Guards).