  We devise an explicit method for computing combinatorial formulae for
Hadamard products of certain rational generating functions. The latter arise
naturally when studying so-called ask zeta functions of direct sums of modules
of matrices or class- and orbit-counting zeta functions of direct products of
nilpotent groups. Our method relies on shuffle compatibility of coloured
permutation statistics and coloured quasisymmetric functions, extending recent
work of Gessel and Zhuang.
