We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding
N
⊴
E
\mathbb {N}\trianglelefteq \mathbb {E}
of locally compact groups and a twisted action
(
α
,
τ
)
(\alpha ,\tau )
thereof on a (post)liminal
C
∗
C^*
-algebra
A
A
the twisted crossed product
A
⋊
α
,
τ
E
A\rtimes _{\alpha ,\tau }\mathbb {E}
is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup
N
⊴
E
\mathbb {N}\trianglelefteq \mathbb {E}
is type-I as soon as
E
\mathbb {E}
is. This happens for instance if
N
\mathbb {N}
is discrete and
E
\mathbb {E}
is Lie, or if
N
\mathbb {N}
is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions.
In the same spirit, call a locally compact group
G
\mathbb {G}
type-I-preserving if all semidirect products
N
⋊
G
\mathbb {N}\rtimes \mathbb {G}
are type-I as soon as
N
\mathbb {N}
is, and linearly type-I-preserving if the same conclusion holds for semidirect products
V
⋊
G
V\rtimes \mathbb {G}
arising from finite-dimensional
G
\mathbb {G}
-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie.