Beretta used several finishes on the 680 series of actions including a painted finish which is like a Duracote finish (682X), the Greystone finish which I think is an electro deposited finish as well as a satin nickel finish. I believe this satin nickel finish is what is on the gun displayed above.
The 680 actions are apparently capable of having a color case hardened finish as is seen on the Coles website. However, I do believe that they are made of an alloy that does not accept traditional bluing well due to the particular alloy - not aluminum. For that reason, I believe that all of the Beretta production guns are either electro deposited with nickel or similar finish or are painted.
I would think that a gunsmith that can due electroless nickel deposition, like on a K80, could do a Beretta 680 series action. If you want to have a case hardened look, then I would try Coles.
Thanks
Z