  For a space X, we define Frobenius and Verschiebung operations on the
nil-terms NA^{fd} (X) in the algebraic K-theory of spaces, in three different
ways. Two applications are included. Firstly, we obtain that the homotopy
groups of NA^{fd} (X) are either trivial or not finitely generated as abelian
groups. Secondly, the Verschiebung defines a Z[N_x]-module structure on the
homotopy groups of NA^{fd} (X), with N_x the multiplicative monoid.
  We also we give a calculation of the homotopy groups of the nil-terms NA^{fd}
(*) after p-completion for an odd prime p as Z_p[N_x]-modules up to dimension
4p-7. We obtain non-trivial groups only in dimension 2p-2, where it is finitely
generated as a Z_p[N_x]-module, and in dimension 2p-1, where it is not finitely
generated as a Z_p[N_x]-module.
