Actions of nilpotent groups on nilpotent groups

For finite nilpotent groups J and N, suppose J acts on N via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow p-subgroups of J that mirrors the primary decomposition of H¹(J, N) for abelian N. We then show that if N ⋊ J acts on some non-empty set Ω, where the action of N is transitive and for each prime p a Sylowp-subgroup of J fixes an element of Ω, then J fixes an element of Ω.