Finite groups whose polynomial permutations are translated power maps

We characterize the finite groups in which every bijective group word with constants has the form x ↦ ax^k b. This is the classification question recorded by Lausch, Nöbauer and Schweiger in 1966 and subsequently discussed as Lausch's problem. A finite 2-group has this property precisely when it has class at most two, or has class three and satisfies an explicit scalar action condition on its derived subgroup together with two specified word identities. In particular, no group of class at least four occurs. The remaining finite groups are obtained from these 2-groups and the known odd-order factors. In the nonnilpotent case an elementary abelian 3-group is acted on through inversion, and an exact condition on the inner power automorphisms of the 2-group determines which extensions occur. The proof combines a polarization argument in class four with a polynomial-permutation obstruction to central 3-extensions. Paper repository: https://github.com/eulogica/finite-group-polynomial-permutations