  In 1997, Cooper-Long established the nontriviality of the kernel of the
modulo 2 Burau representation for the 4-strand braid group, $B_4$. This paper
extends their work by embedding the image group into a finitely presented
group. As an application, we characterize the kernel as the intersection of
$B_4$ with a normal closure of a finite number of elements in an HNN extension
of $B_4$.
