How did the High Priest and rulers of Israel have the authority to extradite Jews from other cities?
The High Priest and rulers of Israel had enough clout with the Roman government to request extradition from certain cities within the Middle Eastern Roman Empire. Judaism carried substantial influence, so those authorities could demand that Jews who had followed Jesus be brought back for judgment.
Les explains that Judaism had a fair amount of clout even with the Romans. The High Priest and rulers of Israel had enough clout with the Roman government that, with a few cities of the Middle Eastern Roman Empire, they could actually demand extradition like we do today. He compares this to America prevailing on Canada to extradite a criminal. The Judaizers, the High Priest of Israel, had that kind of control over the Roman government, so there were certain cities where they could ask for extradition. Saul and the religious leaders of Israel then planned to go to Damascus and bring back renegade Jews who had followed Jesus of Nazareth, and deal with them. Paul later rehearses that they threw them into prison and put them to death.