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We study modular invariance of normalized supercharacters of tame integrablemodules over an affine Lie superalgebra, associated to an arbitrary basic Liesuperalgebra $ \mathfrak{g}. $ For this we develop a several step modificationprocess of multivariable mock theta functions, where at each step a Zwegers'type "modifier" is used. We show that the span of the resulting modifiednormalized supercharacters is $ SL_2(\mathbb{Z}) $-invariant, with thetransformation matrix equal, in the case the Killing form on $\mathfrak{g}$ isnon-degenerate, to that for the subalgebra $ \mathfrak{g}^! $ of $\mathfrak{g}, $ orthogonal to a maximal isotropic set of roots of $\mathfrak{g}. $

Representations of affine superalgebras and mock theta functions III