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In [Indiana Univ. Math. J. 60 (2011), pp. 847-857] the first author introduced second order necessary conditions for a commuting square to admit sequential deformations in the moduli space of non-isomorphic commuting squares. In this paper we investigate these conditions for commuting squares CG constructed from finite groups G. We are especially interested in the case G = Zn, since deformations of CZn correspond to deformations of the Fourier matrix Fn in the moduli space of non-equivalent complex Hadamard matrices. We show that for G = Zn the second order conditions follow automatically from the first order conditions, but this is not necessarily true for other finite abelian groups G. Our result gives a complete description of the second order deformations of the Fourier matrix Fn in the moduli space of non-equivalent complex Hadamard matrices. [ABSTRACT FROM AUTHOR]