For every prime $p\geq5$, we give examples of Beauville $p$-groups whose Beauville structures are never strongly real. This shows that there are purely non-strongly real nilpotent Beauville groups. On the other hand, we determine infinitely many Beauville $2$-groups which are purely strongly real. This answers two questions formulated by Fairbairn in [8].

中文翻译：

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纯（非）强实 Beauville p 群

对于每个素数 $p\geq5$，我们给出了 Beauville $p$-群的例子，这些群的 Beauville 结构从来都不是强实数。这表明存在纯粹的非强实幂零 Beauville 群。另一方面，我们确定了无限多个纯强实的 Beauville $2$-群。这回答了 Fairbairn 在 [8] 中提出的两个问题。