@Drar, As Felipe said, exactly the problem is that $K(t)$ is not as good as $\mathbb{Q}$ for number fields. This is why automorphism group is richer than the Galois group and why it's harder to be computed. I think the use of Hess's automorphism algorithm is to check all the (non-canonical but isomorphic)possible ways of embedding of $F$ in $E$ and check if it's work but I don't know the detail.

I think I got my answer but it rises the new question that what's the logic behind the ordering in small group database. I think I should be able to find my answer if I dig into GAP documents.

Thank you very much. I'm going to check GAP/MAGMA to see if I get the same thing.

Thank you very much for your comment. I'll re-arrange it right away.

@Balazs, I got the book, I'm looking into it to see if I can find there a decent algorithm. @Jack Schmidt: here: [wiki.sagemath.org/magma#GaloisTheory], SageMath people claims that the algorithm is well documented. – Brainy Smurf 1 min ago

I talked to Magma people, their first response was that my "constant field" is too small (in accordance to your solution). They said they'll get back to me. Which haven't happened yet. They claimed that Magma is the only system that has implemented this computation for function field. But I think, for this moment the solving the problem over rational function field (as you did) is enough for me.