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Audaciously kidnapped from a plane in mid-air, Nelson finds himself one of a group of captives on a remote island, waiting to be used as pawn in a plot by Nazis still trying to conquer the world. He's one of several scientists who will be expected to work for the new Fourth Reich! It doesn't take them long to build a radio and signal to Seaview -- but maybe that was exactly what the bad guys expected him to do!
Nelson mentions that the world has been chasing the Nazis for thirty years. This is consistent with the calender in the previous episode Hail to the Chief, which revealed that the series takes place in 1973 - 28 years after the fall of the Third Reich.
Schroder calls himself "Standartenführer". This, along with his uniform (and specifically the sinister Totenkopf, or Death's Head, cap insignia) reveal that he was a member of the Schutzstaffel, or SS. This paramilitary organization was essentially Hitler's private army and was responsible for the bulk of World War II atrocities.
It seems that every television series made in this era had to have at least one episode featuring Nazi villains. Even Star Trek did it. Of course, in 1965 the horrors of that war and of Nazism were much fresher. And the world was still chasing down Nazis who'd fled the country once it became apparent that the Third Reich wasn't going to last anything like as long as Hitler predicted, making this type of story somewhat more credible. As such stories go, this one held together better than most, with a chillingly believable villain and a plot that, while hackneyed now (it's been done twice by the James Bond franchise, for example) was fresh then.