  Let $K$ be a field and $f:\mathbb{P}^N \to \mathbb{P}^N$ a morphism. There is
a natural conjugation action on the space of such morphisms by elements of the
projective linear group $\text{PGL}_{N+1}$. The group of automorphisms, or
stabilizer group, of a given $f$ for this action is known to be a finite group.
In this article, we address two mainly computational problems concerning
automorphism groups. Given a finite subgroup of $\text{PGL}_{N+1}$ determine
endomorphisms of $\mathbb{P}^N$ with that group as subgroup of its automorphism
group. In particular, we show that every finite subgroup occurs infinitely
often and discuss some associated rationality problems. Inversely, given an
endomorphism determine its automorphism group. In particular, we extended the
Faber-Manes-Viray fixed-point algorithm for $\mathbb{P}^1$ to endomorphisms of
$\mathbb{P}^2$. A key component is an explicit bound on the size of the
automorphism group depending on the degree of the endomorphism.
