$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PGL{PGL}$In “Automorphisms of finite linear groups”, Steinberg proves that any automorphism of a simple group of Lie type (normal or twisted) is a product of inner-, diagonal-, field-, and graph automorphisms.

Let $G$ be a simple group of Lie type. Denote by $\hat{G}$ the group of automorphisms generated by $G$ and the diagonal automorphisms of $G$, by $\hat{A}$ the group of automorphism generated by $\hat{G}$ and the field automorphisms of $G$, and by $A$ the whole group of automorphisms of $G$.

As Steinberg explains (3.3-3.6 in his paper), we know that the quotient $\hat{G}/G$ is cyclic (there is one exception), the quotient $\hat{A}/\hat{G}$ is cyclic, and the quotient $A/\hat{A}$ is either trivial or has order $2$ or $6$.

All above, I’m fine with. What puzzles me in Steinberg’s work is the order of the quotient $\hat{G}/G$. He says that $\hat{G}/G$ has order $(n+1,q-1)$, $(2,q-1)$, $(2,q-1)$, $(4,q^n -1)$, $(3,q-1)$, $(2,q-1)$, $(n+1,q+1)$, $(4,q^n+1)$ or $(3,q+1)$ for the respective group $A_n$, $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $A_n^1$, $D_n^1$ or $E_6^1$. I do not understand what he means by this. To my understanding the first number in the brackets is the order of the quotient $\hat{G}/G$ (e.g. in $A_1=\PSL_2(q)$ we have $\hat{G}/G=\PGL_2(F)/{\PSL_2(F)}$ which has order $2=n+1$ when $q$ is odd) but I don’t then understand what the second number in the brackets stands for (e.g. in $A_1=\PSL_2(q)$, what is $q-1$ for? It is the order of the maximal split torus but how is this related to the order of the group $\PGL_2(F)/{\PSL_2(F)}$)?

My second question is about the automorphism groups of Suzuki and Ree groups. They do not appear in Steinberg's work. Is there a similar description for those automorphism groups and where could I find it?