  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~$2$).
