This paper links the third symmetric cohomology (introduced by Staic and Zarelua ) to crossed modules with certain properties. The equivalent result in the language of 2-groups states that an extension of 2-groups corresponds to an element of $HS^3$ iff it possesses a section which preserves inverses in the 2-categorical sense. This ties in with Staic's (and Zarelua's) result regarding $HS^2$ and abelian extensions of groups.

中文翻译：

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群的交叉模和对称上同调

本文将第三个对称上同调（由 Staic 和 Zarelua 引入）与具有某些性质的交叉模块联系起来。2-groups 语言中的等效结果表明，2-groups 的扩展对应于 $HS^3$ 的元素，如果它具有保留 2-categorical 意义上的逆的部分。这与 Staic（和 Zarelua）关于 $HS^2$ 和群的阿贝尔扩展的结果有关。