  In this paper we find the largest automorphism group of a smooth cubic
surface over any finite field of characteristic $2.$ We prove that if the order
of the field is a power of $4,$ then the automorphism group of maximal order
of~a~smooth cubic surface over this field is $\mathrm{PSU}_4(\mathbb{F}_2).$ If
the order of the field of characteristic $2$ is not a power of $4,$ then we
prove that the automorphism group of maximal order of a smooth cubic surface
over this field is the symmetric group of degree $6.$ Moreover, we prove that
smooth cubic surfaces with such properties are unique up to isomorphism.
