The character table of the finite Chevalley group $F_4(q)$ for $q$ a power of~$2$

Let $q$ be a prime power and $F_4(q)$ be the Chevalley group of type $F_4$ over a finite field with $q$ elements. Marcelo–Shinoda (1995) determined the values of the unipotent characters of $F_4(q)$ on all unipotent elements, extending earlier work by Kawanaka and Lusztig to small characteristics. Assuming that $q$ is a power of~$2$, we explain how to construct the complete character table of~$F_4(q)$.