I found the following statement in Cameron's Projective and Polar Spaces (page 120 here):
If two such automorphisms of order $3$ have a common fixed line, then they generate a $\{2,3\}$-group, since the stabiliser of a line in $GL(4,2)$ is a $\{2,3\}$-group.
What does $\{2,3\}$-group mean here? It doesn't seem to be defined anywhere else in the book, and I was unable to find a definition online. The discussion is related to alternating groups, if that helps.
It means a finite group such that the only primes dividing its order are $2$ and $3$.