  In this paper we present a description of the Galois representation attached
to an elliptic curve defined over a $2$-adic field $K$, in the case where the
image of inertia is non-abelian. There are two possibilities for the image of
inertia, namely $Q_8$ and $SL_2(\mathbb{F}_3)$, and in each case we need to
distinguish whether the inertia degree of $K$ over $\mathbb{Q}_2$ is even or
odd. The result presented here can be implemented in an algorithm to compute
explicitly the Galois representation in these four cases.
