From my understanding, the discussion of Jesus being betrayed 'by the Jews' was a later addition in order to survive under a Roman Empire. The guards may be more inclined to let them go just this once if they loved the Roman Empire and believed they were just forced into it 'by the Jews'.
Contrary, if it was that the Roman Empire who killed their God incarnate, people under Roman (and of Roman descent) rule would be less inclined to listen (guilt by association), and authorities would be far less inclined to let them go when they would have been seen as, effectively, terrorists with a grudge to attempt to destroy the empire. As Christianity became more related to the gentiles and less on the Jewish people, the Jewish people became more demonized.
I don't believe the Jewish people killed Jesus nor attempted to, at all. The Romans killed him, in fact there is every chance that the Jewish High Priest (Yosef bar Kayafa, Caiaphas) tried to dissuade Pontius from killing Jesus, by pointing out Jesus was just a harmless madman. The Jewish people were his people, his friends, his family. Why would they give him over to their oppressors because of his claims?
The demonising of the Jewish people was needed to survive in a hostile, alien nation. This became one of the major problems of Christianity in days long gone, and to some extent, still today.
That's my understanding of it, anyway.
