Permutation groups of prime power degree and $p$-complements

Extending earlier work of Guralnick and of Cai and Zhang, we classify the almost simple groups which have transitive permutation representations of prime power degree $p^k$, and those which have $p$-complements (stabilisers of order coprime to $p$ in such representations). We deduce that every primitive permutation group of prime power degree has a regular subgroup, and that any two faithful primitive representations of a group, of the same prime power degree, are equivalent under automorphisms. In general, $p$-complements in a finite group can be inequivalent under automorphisms, or even non-isomorphic. We extend examples of such phenomena due to Buturlakin, Revin and Nesterov by showing that the number of inequivalent classes of complements can be arbitrarily large. Questions concerning the existence of prime power representations and $p$-complements in groups with socle ${\rm PSL}_d(q)$ are related to some difficult open problems in Number Theory.