Rome ruled Palestine ( their name for Israael, BTW ) and the Jews had absolutely NO say so over any of the laws. Jesus said " RENDER UNTO CAESAR ...", because Caesar WAS Rome and the head of the empire. The ONLY person, who could render the sentence of crucifixtion, was Pontius Pilate.
The Ancient Roman Empire was pretty easy about the religious practices, the peoples in the places they ruled over followed, with one caveat. They demanded that those peoples also acept their Caesars as gods and follow their laws. The Jews, being the oNLY monotheists, threw a HUGE monkey wrench into that mix, as they refused to accept any god but GOD; hence a lot of the friction.
During and even predating Jesus' time on earth, there were various sects within Judaism; some more militant than others. This was a problem for the Romans, since the Jewish Messiah, was , acording to Jewish traditions, supposed to come to fight . There were many " leaders ", at this time, doing just that ... fighting the Romans. They had to be put down.
But, the Jews had NO power, none whatsoever, in telling Pilate who or how to kill anyone.
The fact remains, according to the Bible they had the Romans execute Jesus..according to THEIR religious law. The Romans had no interest in executing Jesus.
Pilate found no fault in Jesus and washed his hands of it..while the Jews gladly accepted responsibility for his execution..and said "let his blood be on us".