Let $G$ be a finite nilpotent nonabelian group. Is it true that for every natural number $k$ there exists a finite group $G_k$ such $G_k$ is not isomorphic to a subgroup of a direct power of $G$ while every $k$-generated subgroup of $G_k$ is isomorphic to such a subgroup.
I know that for abelian groups this is not possible.
This is true and has been proved in Olʹšanskiĭ, A. Ju. Conditional identities in finite groups. Sibirsk. Mat. Ž. 15 (1974), 1409–1413. In fact he proved a similar fact for any finite group with a non-abelian Sylow subgroup. Moreover if all Sylow subgroups are abelian, the fact is no longer true.