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In [ABB] Asmar, Berkson and Bourgain prove that for a sequence ${{\varphi }_j}_{j=1}^{\infty }$ of weak type (1, 1) multipliers in ${ℝ}^n$ and a function $k\in L^1\left({ℝ}^n\right)$ the weak type (1,1) constant of the maximal operator associated with ${k⁎{\varphi }_j}_j$ is controlled by that of the maximal operator associated with ${{\varphi }_j}_j$. In [ABG] this theorem is extended to LCA groups with an extra hypothesis: the multipliers must be continuous. In this paper we prove a more general version of this last result without assuming the continuity of the multipliers. The proof arises...