Homotopy comultiplications on the localization of a wedge of spheres and Moore spaces

For an abelian group $ G $ and a positive integer $ n $, we set $ M(G, n) $ as a Moore space of type $ (G, n) $. In this paper, for a prime number $ p $, we are interested in the structure of homotopy comultiplications on the localization $ L_{(p)} $ of a wedge $ L: = \mathbb S^m \vee M(G, n) $ of the homotopy spheres and the Moore spaces for $ 2 \leq m < n $. We also provide a list of examples to examine the phenomena of homotopy comultiplications on $ L_{(p)} $.