/auto/ is an abbreviation for /automobile/.
/ auto- / is a prefix with Greek origin (auton) the meaning of /self/ or /by itself/, /by himself/.
The word /auto/ does not exist (as far as I know) in Italian;
examples, whose meaning could easily be found on a dictionary:
autonomo (adj.), autoportante, autobus (Greek+Latin here!),
autorita` (I don't think has the same auto- prefix, just looks like),
autonomo (Greek + Latin) : has its own laws, therefore: does not depend on somebody
automatico : thinks by itself, therefore: does not need an operator (Greek + indoeuropean /men/: mind)
I studied a little bit of Greek for one year and always had poor grades. Thank you for this question, I am trying to redeem autonomously.