  We study the groups of automorphisms of rational algebraic surfaces that
admit a relatively minimal pencil of curves of arithmetic genus one over an
algebraically closed field of arbitrary characteristic. In particular, we
classify such surfaces that admit non-trivial automorphisms that act trivially
on the Picard group. As an application, we classify classical Enriques surfaces
in characteristic $2$ that admit non-trivial numerically trivial automorphisms.
