  In this paper we give an explicit description of the automorphism group of a
primary Kodaira surface $X$ in terms of suitable liftings to the universal
cover $\mathbb{C}^2$. As it happens for complex tori, the automorphism group of
$X$ is an extension of a finite cyclic group related to the action of the
automorphisms on the tangent bundle of $X$ by a group of automorphisms which
acts on $X$ as translations (both on the base and on the fibres of the elliptic
fibraion of $X$). We can then characterize those automorphisms acting
symplectically and those acting trivially on cohomology. Finally, we describe
the fixed locus of an automorphism and we show that, if not empty, it is the
union of a finite number of fibres of the elliptic fibration of $X$. As a
byproduct, we provide a different proof of Borcea's description of the moduli
space of Kodaira surfaces.
