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
$\mathbb{Q}[G]$
.