In this paper we study the saturated fusion systems over a direct product of the extraspecial group of order p^3 of exponent p and a finite abelian p-group. The result provides some new exotic fusion systems, whose unique components are isomorphic to the exotic fusion systems over 7+^1+2 found by Ruiz and Viruel.
Let G be a polycyclic group and α a regular automorphism of order four of G. If the map φ: G→ G defined by g~φ= [g, α] is surjective, then the second derived group of G is contained in the centre of G. Abandoning the condition on surjectivity, we prove that C_G(α~2) and G/[G, α~2] are both abelian-by-finite.
