  It follows from an observation of A. Coble in 1919 that the automorphism
group of an unnodal Enriques surface contains the $2$-congruence subgroup of
the Weyl group of the $E_{10}$-lattice. In this article, we determine how much
bigger the automorphism group of an unnodal Enriques surface can be.
Furthermore, we show that the automorphism group is in fact equal to the
$2$-congruence subgroup for generic Enriques surfaces in arbitrary
characteristic (under the additional assumption that the Enriques surface is
ordinary if the characteristic is $2$), improving the corresponding result of
W. Barth and C. Peters for very general Enriques surfaces over the complex
numbers.
