This is a follow up question to Automorphisms of a character table
Consider the character table of a finite group $ G $. Every outer automorphism of $ G $ permutes the characters of $ G $. If some of the characters are non-integral then we can consider the corresponding Galois automorphisms and they will also permute the characters of $ G $.
If $ G $ is a finite quasisimple group do outer automorphisms and Galois automorphisms together generate all possible automorphisms of the character table of $ G $? Is there a name for the group formed by the outer automorphisms and the Galois automorphisms?
From my experience when $ G $ is quasisimple these seem to account for all the table automorphisms.
For example:
$ SL(2,5) $ --- There is only one (non-trivial) automorphism of the character table of $ SL(2,5) $. It swaps the two degree $ 2 $ irreps and swaps the two degree $ 3 $ irreps. All characters of $ SL(2,5) $ are defined over $ \mathbb{Q}(\sqrt{5}) $. The Galois automorphism $ \sqrt{5} \to -\sqrt{5} $ enacts exactly the table automorphism described above. There is also a unique nontrivial outer automorphism of $ SL(2,5) $ and it also enacts exactly the table automorphism described above. So in this case the Galois automorphisms and the outer automorphisms overlap, corresponding to the same table automorphism.
$ 3.A_6 $--- This group has outer automorphism group $ 2^2 $. All characters of $ 3.A_6 $ are defined over $ \mathbb{Q}(\sqrt{5},\sqrt{-3}) $, so the group of Galois automorphisms is also a $ 2^2 $. However in this case these groups do not totally overlap and they generate $ 2^3 $ different table automorphisms. In particular there are two independent table automorphisms coming from Galois conjugation and then there is also a third independent table automorphism coming from an outer automorphism (this table automorphism swaps the two degree $ 5 $ irreps, which are both integral, lifted from $ A_6 $).