On the full automorphism groups of $2$-designs constructed from finite fields ${\mathbb{F}}_{2^n}$

In this manuscript, for $q:=2^n$ with $n\ge 2$, we study two primitive maximal subgroups of the alternating group ${\mathsf{A}}_{q-1}$. These subgroups are the full automorphism groups of $2$-designs which are constructed from algebraic curves over the finite field ${\mathbb{F}}_q$.