An infinite family of 2-groups with mixed Beauville structures

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We construct an infinite family of triples (Gₖ,Hₖ,Tₖ), where Gₖ are 2-groups of increasing order, Hₖ are index-2 subgroups of Gₖ, and Tₖ are pairs of generators of Hₖ. We show that the triples uₖ = (Gₖ,Hₖ,Tₖ) are mixed Beauville structures if k is not a power of 2. This is the first known infinite family of 2-groups admitting mixed Beauville structures. Moreover, the associated Beauville surface S(u₃) is real and, for k > 3 not a power of 2, the Beauville surface S(uₖ) is not biholomorphic to S(uₖ).