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$\mathbb{Q}$
We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions 
$K/\mathbb{Q}$ with Galois group isomorphic to 
$A_4$, 
$S_4$, 
$A_5$, and dihedral groups of order 
$2p^n$ for certain prime powers 
$p^n$. In particular, when 
$K/\mathbb{Q}$ is a Galois extension with Galois group 
$G$ isomorphic to 
$A_4$, 
$S_4$ or 
$A_5$, we give necessary and sufficient conditions for the ring of integers 
$\mathcal{O}_{K}$ to be free over its associated order in the rational group algebra 
$\mathbb{Q}[G]$.
