Hopf Galois structures, skew braces for groups of size pnq : The cyclic Sylow subgroup case

Let n >= 1 be an integer, p, q be distinct odd primes. Let G, N be two groups of order p(n)q with their Sylow-p-subgroups being cyclic. We enumerate the Hopf-Galois structures on a Galois G-extension, with type N . This also computes the number of skew braces with additive group isomorphic to N and multiplicative group isomorphic to G . Further when q < p , we give a complete classification of the Hopf-Galois structures on Galois-G-extensions.