Let n >= 1 be an integer, p, q be distinct odd primes. Let G, N be two groups of order pⁿq 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.

The first named author is partially supported by the IMSc postdoctoral fellowship and the second author has been partially supported by the HRI postdoctoral fellowship.