This thesis contains a series of studies about 2-adic integral Galois representations unramified outside a finite set of primes. There are two main focuses of research: the study of 2-adic integral Galois representations and the study on how to compare two 2-adic integral Galois representations. Firstly, when studying a representation, we develop methods to determine whether the residual image is reducible or irreducible: in the irreducible case the residual image is completely determined. On the other hand, when the residual image is reducible we are able to make a choice of a stable lattice to completely determine the residual image. Lastly, from the choice of lattice, we are able to extend our methods to determine whether the representation is trivial modulo 2k+1 assuming that is trivial modulo 2k. Secondly, when comparing two 2-adic integral Galois representations, we are able to determine whether the representations are isogenous that is, after conjugation if necessary, their residual representations are the same. In some cases, this process follows the approach given in [11] by Ron Livné. Finally, the idea behind these studies was the notion of what we call a "Black Box representation", i.e., a system that will provide the characteristic polynomial of the representation for any prime not in the set of primes.