<p>Kajiura et al. in (J Algebraic Combin 58:113–135, 2023) studied an approximation of the integration of a function over a finite group <i>G</i> by a subset <i>Y</i> of <i>G</i> as an analogue of quasi-Monte Carlo (QMC) methods over hypercubes. In particular, they gave a Koksma-Hlawka type upper bound on the integration error and a lower bound on the term depending only on <i>Y</i> in the error-bound. Furthermore, they proved that the lower bound is achieved if and only if <i>Y</i> is a pre-difference set, which is a generalization of a difference set in combinatorial design theory. In this paper, we give an upper bound on the integration error slightly improving the error-bound given by Kajiura–Matsumoto–Okuda by using a normal subgroup of <i>G</i>. Furthermore, we give lower bounds on the terms depending only on <i>Y</i> in the improved error-bound. We introduce the concept of divisible pre-difference sets as a generalization of divisible difference sets, which have been studied in combinatorial design theory. Then, we prove that the lower bounds are achieved if and only if <i>Y</i> is a divisible pre-difference set. We also give some constructions of divisible pre-difference sets and enumerate divisible pre-difference sets of small order.</p>