  For a p-group G admitting an automorphism $\phi$ of order $p^n$ with exactly
$p^m$ fixed points such that $\phi^{p^{n-1}}$ has exactly $p^k$ fixed points,
we prove that G has a fully-invariant subgroup of m-bounded nilpotency class
with $(p,n,m,k)$-bounded index in G. We also establish its analogue for Lie
p-rings. The proofs make use of the theory of commutator-type operators.
