Each automorphism is determined entirely by where it sends one of the primitives. The fixed field is the same.

In your case just look to see the possibilities.

So you are correct in that it has order 4. It is pretty easy to see which one it is because there are only 2 groups of order 4. Klein 4 is characterized by the fact that every element has order 2. Cyclic 4 has some elements of order 4.

Check the order of these automorphisms (start with a) and see what you get.