An infinite family of 2-groups with mixed Beauville structures

We construct an infinite family of triples $(G_k,H_k,T_k)$, where $G_k$ are 2-groups of increasing order, $H_k$ are index-2 subgroups of $G_k$, and $T_k$ are pairs of generators of $H_k$. We show that the triples $u_k = (G_k,H_k,T_k)$ 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 $S(u_3)$ is real and, for $k > 3$ not a power of 2, the Beauville surface $S(u_k)$ is not biholomorphic to $\bar{S(u_k)}$.