Wild Galois representations: elliptic curves over a $2$-adic field with non-abelian inertia action

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 $S{L}_2({\mathbb{F}}_3)$, 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.