This expression refers to people exercising authority over the disciples. AT: "looked for a way to arrest Jesus" (See: [[rc://en/ta/man/translate/figs-metonymy]])
This is the reason that they did not arrest Jesus right away. The people respected Jesus, and the religious leaders were afraid of what the people might do if they arrested him. AT: "they did not arrest him because they were afraid of the people" (See: [[rc://en/ta/man/translate/figs-explicit]])
"Rule" and "authority" are two ways of saying that they wanted the governor to judge Jesus. It can be translated with one or both expressions. AT: "so that the governor would punish Jesus" (See: [[rc://en/ta/man/translate/figs-explicit]])