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For any algebraically closed field $K$ and any endomorphism $f$ of$\mathbb{P}^1(K)$ of degree at least 2, the automorphisms of $f$ are theM\"obius transformations that commute with $f$, and these form a finitesubgroup of $\operatorname{PGL}_2(K)$. In the moduli space of complex dynamicalsystems, the locus of maps with nontrivial automorphisms has been studied indetail and there are techniques for constructing maps with prescribedautomorphism groups that date back to Klein. We study the correspondingquestions when $K$ is the algebraic closure $\bar{\mathbb{F}}_p$ of a finitefield. We use the classification of finite subgroups of$\operatorname{PGL}_2(\bar{\mathbb{F}}_p)$ to show that every finite subgroupis realizable as an automorphism group. To construct examples, we use methodsfrom modular invariant theory. Then, we calculate the locus of maps over$\bar{\mathbb{F}}_p$ of degree $2$ with nontrivial automorphisms, showing howthe geometry and possible automorphism groups depend on the prime $p$.

Automorphism Groups of Endomorphisms of $\mathbb{P}^1 (\bar{\mathbb{F}}_p)$