4 problems

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, and let $\mathcal{C}$ be a finite category. If $\mathcal{C}$ is a fin…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, and let $\mathcal{C}$ be a finite category. Let $\operatorname{Aut}(\mathcal{C})$ denote its automorphism gro…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, and let $\mathcal{C}$ be a finite EI-category, meaning that every end…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, and let $\mathcal{C}$ be a finite category with $p$-orbit category…