Let p, q be distinct primes, with p > 2. In a previous paper we classified the Hopf-Galois structures on Galois extensions of degree p²q, when the Sylow p-subgroups of the Galois group are cyclic. This is equivalent to classifying the skew braces of order p²q, for which the Sylow p-subgroups of the multiplicative group are cyclic. In this paper we complete the classification by dealing with the case when the Sylow p-subgroups of the Galois group are elementary abelian.

According to Greither and Pareigis, and Byott, we will do this by classifying, for the groups (G, .) of order p²q, the regular subgroups of their holomorphs whose Sylow p-subgroups are elementary abelian.

We rely on the use of certain gamma functions γ:G -> Aut(G). These functions are in one-to-one correspondence with the regular subgroups of the holomorph of G, and are characterised by the functional equation γ(g^γ(h) . h) = γ(g) γ(h), for g, h in G. We develop methods to deal with these functions, with the aim of making their enumeration easier and more conceptual.

This paper is part of the thesis submitted by E. Campedel as a requirement for the PhD degree at the University of Milano--Bicocca. The authors are members of INdAM---GNSAGA. The authors gratefully acknowledge support from the Departments of Mathematics of the Universities of Milano--Bicocca, Pisa, and Trento.