  The modular properties of characters of representations of a family of
W-superalgebras extending the affine Lie superalgebra of gl(1|1) are
considered. Modules fall into two classes, the generic type and the non-generic
one. Characters of non-generic modules are expressed in terms of higher-level
Appell-Lerch sums. We compute the modular transformations of characters and
interpret the Mordell integral as an integral over characters of generic
representations. The \C-span of a finite number of non-generic characters
together with an uncountable set of characters of the generic type combine into
a representation of SL(2;\Z). The modular transformations are then used to
define a product on the space of characters. The fusion rules of the extended
algebras are partially inherited from the known fusion rules for modules of the
affine Lie superalgebra of gl(1|1). Moreover, the product obtained from the
modular transformations coincides with the product of the Grothendieck ring of
characters if and only if the fusion multiplicities are at most one.
