Do you think it is Sauer an Sohn?
Did you see this posted by Clay?
"I’ve seen the SS marking attributed to barrel maker Stephan Schilling in the past. Barrels get kind of convoluted with steel suppliers, blank providers, and finishers etc."
Clay and Fal Grunt are two of the most knowledgeable guys on this board when it comes to commercial guns (but certainly not the only ones)...this is a commercial gun, as pointed out by Fal Grunt..."Behind the recoil lug, the 10197 serial number tells us the "action" is a Mauser Oberndorf manufactured action, sold to the trade." Sold to trade means a gunsmith or gunmaker bought the action from MO to build the rifle, the barrel might be Stephan Schilling, or another barrel supplier using S S as their mark.
"The barrel 172/28 tells us the gun was assembled and proofed pre ~ 1912." Pre 1912 there was no hint of the SS, not even in a dream...pre 1912 is well before WWI even so this has no connection to anything made purposefully for any organization post WWI. Whether somehow this commercial hunting gun found its way into some sort of Police stores somehow, no one can say, very doubtful, and there is no markings that would lead one to that direction...
It is what it is, a nice commercial trade made gun...someone's high end hunting gun. Enjoy it for what it is, don't try to grasp straws to make it what it is not...