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Let $p, q$ be distinct primes, with $p > 2$. In a previous paper weclassified the Hopf-Galois structures on Galois extensions of degree $p^{2} q$,when the Sylow $p$-subgroups of the Galois group are cyclic. This is equivalentto classifying the skew braces of order $p^2q$, for which the Sylow$p$-subgroups of the multiplicative group is cyclic. In this paper we completethe classification by dealing with the case when the Sylow $p$-subgroups of theGalois group are elementary abelian.  According to Greither and Pareigis, and Byott, we will do this by classifyingthe regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order$p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are elementaryabelian.  We rely on the use of certain gamma functions $\gamma:G\to\operatorname{Aut}(G)$. These functions are in one-to-one correspondence withthe regular subgroups of the holomorph of $G$, and are characterised by thefunctional equation $\gamma(g^{\gamma(h)} \cdot h) = \gamma(g) \gamma(h)$, for$g, h \in G$. We develop methods to deal with these functions, with the aim ofmaking their enumeration easier and more conceptual.

Hopf-Galois structures on extensions of degree $p^{2} q$ and skew braces of order $p^{2} q$: the elementary abelian Sylow $p$-subgroup case