Series of $p$-groups with Beauville structure

For every $p\ge 2$ we show that each finite $p$-group with an unmixed Beauville structure is part of a surjective infinite projective system of finite $p$-groups with compatible unmixed Beauville structures. This leads to the new notion of an unmixed topological Beauville structure on pro-finite groups. We further construct for $p\ge 5$ a new explicit infinite series of non-abelian $p$-groups that allow unmixed Beauville structures.