  We generalize the Donagi and Witten construction of a first obstruction class
for splitting of a supermanifold via differential operators using the theory of
$n$-fold vector bundles and graded manifolds. Applying the generalized
Donagi--Witten construction we obtain a family of embeddings of the category of
supermanifolds into the category of $n$-fold vector bundles and into the
category of graded manifolds. This leads to a realization of any non-split
supermanifold in terms of a collection of vector bundles and some morphism
between them. Further we study the images of these embeddings into the category
of graded manifolds in the case of a Lie supergroup and a Lie superalgebra. We
show that these images satisfy universal property of a graded covering or a
graded semicovering.
