  On a group $G$, a filtration by normal subgroups is referred to as a normal
series. If subsequent quotients are abelian, the filtration is referred to as
an \emph{abelian-quotient normal series}, or `AQ normal series' for short. In
this article we consider `sheaves of AQ normal series'. From a given AQ normal
series satisfying an additional hypothesis we derive a complex whose first
cohomology obstructs the resolution of an `integration problem'. These
constructs are then applied to the classification of supermanifolds modelled on
$(X, T^*_{X, -})$, where $X$ is a complex manifold and $T^*_{X, -}$ is a
holomorphic vector bundle. We are lead to the notion of an `obstruction
complex' associated to a model $(X, T^*_{X, -})$ whose cohomology is referred
to as `obstruction cohomology'. We deduce a number of interesting consequences
of a vanishing first obstruction cohomology. Among the more interesting
consequences are its relation to projectability of supermanifolds and a
`Batchelor-type' theorem: if the obstruction cohomology of a `good' model $(X,
T^*_{X, -})$ vanishes, then any supermanifold modelled on $(X, T^*_{X, -})$
will be split.
