In the book On the Automorphisms of the Classical Groups, J. Dieudonné the author describes all the automorphisms of $\text{GL}(n,\mathbb{F}_q)$, but I couldn't find a description of the automorphism group as a group, for example what is it isomorphic to or what is the cardinality of the group. Can someone throw some light on the mentioned questions?
Thanks in advance.