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The only thing the SS option bought you with an Impala was bucket seats and chrome trim. It was not a performance option on the B-body cars (Impala, Caprice, Bel Air, etc.) the way it was on the 1968 F-body Camaro, or the A-body Chevelle, or the X-body Nova which when an SS was ordered changed the suspension to heavy duty parts (RPO code F-41), added to HD cooling, HD electrical and of course that ever popular "High Performance" engine option. You could order an Impala SS with a 250 cubic inch six cylinder in 1968 if you wanted to.
Some one wanted a sportier car which is why they added the F-41 suspension package (stiffer spring rate and HD gas filled shocks and it included a front sway bar). Though why they chose to install a 396 when only the Corvette or the Impala had as an option 427 is beyond me.
The only "true" SS Impala built in 1968 was the 427 SS (Note that is not an SS with a 427 installed) which was the only way to get the option of the 425 horse 427. It included a special hood only offered with the 427 SS and all of the goodies that Chevrolet added to the other SS models they made (such as HD suspension, HD cooling, HD electrical, etc.) in 1968.
Chevy only offered the rare 427 SS option in 1967, '68, and '69 before it was dropped due to people choosing a lighter car to have a high performance motor installed in.