  We investigate the number and the geometry of smooth hyperelliptic curves on
a general complex abelian surface. We show that the only possibilities of
genera of such curves are $2,3,4$ and $5$. We focus on the genus 5 case. We
prove that up to translation, there is a unique hyperelliptic curve in the
linear system of a general $(1,4)$ polarised abelian surface. Moreover, the
curve is invariant with respect to a subgroup of translations isomorphic to the
Klein group. We give the decomposition of the Jacobian of such a curve into
abelian subvarieties displaying Jacobians of quotient curves and Prym
varieties. Motivated by the construction, we prove the statement: every \'etale
Klein covering of a hyperelliptic curve is a hyperelliptic curve, provided that
the group of $2$-torsion points defining the covering is non-isotropic with
respect to the Weil pairing and every element of this group can be written as a
difference of two Weierstrass points.
