Automorphisms of quartic del Pezzo surfaces in characteristic zero

For each field $k$ of characteristic zero, we classify which groups act by automorphisms on a quartic del Pezzo surface over $k$. We also determine which groups act on $k$-rational, stably $k$-rational, or $k$-unirational quartic del Pezzo surfaces. For each group $G$ that is realized over $k$, we exhibit explicit equations for a quartic del Pezzo surface $X$ in $\mathbb{P}^4_k$ such that $G$ acts by automorphisms on $X$.