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The Steinitz class of a number field extension $K/k$ is an ideal class in the ring of integers ${𝒪}_k$ of $k$, which, together with the degree $[K:k]$ of the extension determines the ${𝒪}_k$-module structure of ${𝒪}_K$. We denote by ${\mathrm{R}}_t(k,G)$ the set of classes which are Steinitz classes of a tamely ramified $G$-extension of $k$. We will say that those classes are realizable for the group $G$; it is conjectured that the set of realizable classes is always a group. In this paper we will develop some of the ideas contained in [...