aŭtomata

Here the aŭto- really is the Greek "self". Aŭtomata means acting of its own accord: nobody has to push it. The prefix is genuine and productive in international vocabulary - aŭtobiografio, aŭtomobilo, aŭtonoma - and this is one of the words where a learner's guess pays off.

Which is precisely why the trap next door matters. Aŭtoritato has nothing to do with it: that word is Latin all the way down, from auctor, and its aŭ is a coincidence of spelling. Meanwhile aŭto, the car, owns the letters only by abbreviation - it is a clipping of aŭtomobilo, which is where the Greek self actually lives. Three separate things, one spelling, and the pronunciation gives no help because aŭ is a single diphthong in all of them.

Use aŭtomate for the adverb, and note the everyday extension that English shares: a reaction can be aŭtomata without any machinery involved, meaning you did it without thinking. Aŭtomatigi is to automate something, and aŭtomato on its own is the machine - a vending machine, a cash machine, an automaton.