Let $E$ be an elliptic curve over $\mathbb{Q}$. For $\ell$-adic representations associated to $E$, much is understood about the sizes of the images and the conjugacy invariants of the images of Frobenius elements. On the other hand, much less is known about the outer Galois representations associated to $E$. These are representations from $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to an outer automorphism group of a free pro-$\ell$ group. \newline The goal of this thesis is to take a first step in understanding more concretely more general Galois representations associated to $E$ (to the automorphism group of a non-abelian group). In particular, I study the surjectivity of a Galois representation to a certain subgroup of the automorphism group of a metabelian group. I use the Frattini lifting theorem to turn the question of the surjectivity of this representation into a question about the surjectivity of a representation to the Frattini quotient of this group. Then, I compute some conjugacy invariants for the images of the Frobenius elements. This will give rise to new arithmetic information analogous to traces of Frobenius of the $\ell$-adic representation.