The cover-avoidance property and its variations have become central in the study of finite groups. Along with this widespread attention comes a need to construct examples and counterexamples. The main goal of this work is to apply the theory of direct products to systematically facilitate such constructions.

中文翻译：

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覆盖回避特性及其变化已成为有限群研究的核心。随着这种广泛关注，需要构建示例和反例。这项工作的主要目标是应用直接积的理论来系统地促进这种结构。