  Let $G$ be a finite group of Lie type. In order to determine the character
table of $G$, Lusztig developed the theory of character sheaves. In this
framework, one has to find the transformation between two bases for the space
of class functions on $G$, one of them being the irreducible characters of $G$,
the other one consisting of characteristic functions associated to character
sheaves. In principle, this has been achieved by Lusztig and Shoji, but the
underlying process involves some scalars which are still unknown in many cases.
The problem of specifying these scalars can be reduced to considering cuspidal
character sheaves. We will deal with the latter for the specific case where
$G=E_7(q)$, and $q$ is a power of the bad prime $p=2$ for $E_7$.
