A recently proposed technique allows one to constrain both the background and perturbation cosmological parameters through the distribution function of supernova Ia apparent magnitudes. Here we extend this technique to alternative cosmological scenarios, in which the growth of structure does not follow the $Lambda$CDM prescription. We apply the method first to the supernova data provided by the JLA catalog combined with all the current independent redshift distortion data and with low-redshift cluster data from Chandra and show that although the supernovae alone are not very constraining, they help in reducing the confidence regions. Then we apply our method to future data from LSST and from a survey that approximates the Euclid satellite mission. In this case we show that the combined data are nicely complementary and can constrain the normalization $sigma_8$ and the growth rate index $gamma$ to within $0.6%$ and $7%$, respectively. In particular, the LSST supernova catalog is forecast to give the constraint $gamma (sigma_8/0.83)^{6.7} = 0.55 pm 0.1$. We also report on constraints relative to a step-wise parametrization of the growth rate of structures. These results show that supernova lensing serves as a good cross-check on the measurement of perturbation parameters from more standard techniques.
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