On the character tables of the finite reductive groups $E_6(q)_{\text{ad}}$ and ${^2\!E}_6(q)_{\text{ad}}$

We show how the character tables of the groups $E_6(q)_{\text{ad}}$ and ${^2\!E}_6(q)_{\text{ad}}$ can be constructed, where $q$ is a power of~$2$. (Partial results are also obtained for any $q$ not divisible by~$3$.) This is based on previous work by Hetz, Lusztig, Malle, Mizuno and Shoji, plus computations using Michel's version of {\sf CHEVIE}. We also need some general results that are specific to semisimple groups which are not of simply connected type. A further crucial ingredient is the determination of the values of the unipotent characters on unipotent elements for groups of type $D_4$ and $D_5$ (in characteristic~$2$).