  For every $p\geq 2$ 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 $p \geq 5$ a new explicit infinite series of
non-abelian $p$-groups that allow unmixed Beauville structures.
