I'm quite interested in this myself. I'll try to answer, in hope that someone else can complete it if I'm missing something important.
The only two major cases of automorphic induction known still are:
Local fields (Henniart-Herb)
Cyclic Galois extension of prime degree (Arthur-Clozel)
For a recent source that only mentions these two examples, see Colin Bushnell's paper for the 2014 book "Automorphic Forms and Galois Representations, Volumen 1".
Some other known instances of automorphic induction include:
Non-normal cubic extension (Jacquet-Piatetski-Shapiro-Shalika)
Non-normal extensions with solvable Galois closure for certain Hecke characters (Harris)
A case of non-normal quintic extension with non-solvable closure (Kim)
Feel free to comment or edit if you have knowledge of any other relevant result.