I’ve worked on a couple Empire rifles and they are a high quality action.
Given that there is a bit of rust showing, I would remove the shroud assembly before soaking it in Kroil. You can do so by holding the cocking piece sides in a vise and pulling the bolt body forward while unscrewing the shroud. If you don’t have smooth vise jaws line them with brass or aluminum to prevent marring the cocking piece.

"… you may all go to hell, and I will go to Texas.” Davy Crockett