  Let $k$ be an algebraically closed field, $S$ a variety over $k$ and m a
nonnegative integer. There is a space $S_m$ over $S$ , called the jet scheme of
$X$ of order $m$, parameterizing $m$-th jets on $S$. The fiber over the
singular locus of $S$ is called the singular fiber. In this paper, we consider
the singular fibers of the jet schemes of 2-dimensional rational double points
over a field $k$ of characteristic $2$ whose resolution graph is of type $D_4$.
There are two types of such singularities, of type $D_4^0$ and type $D_4^1$. We
give the irreducible decomposition of the singular fiber and describe the
configuration of the irreducible components. The case of a $D_4^0$-singularity
is quite similar to the case of characteristic $0$ studied in [3]. The case of
$D_4^1$-singularity requires more elaborate analysis of certain subsets of the
singular fiber.
