  In this article we consider Riemann surfaces and abelian varieties endowed
with a group of automorphisms isomorphic to a generalized quaternion group. We
provide isogeny decompositions of each abelian variety with this action,
compute dimensions of the corresponding factors and provide conditions under
which this decomposition is nontrivial. We then specialize our results to the
case of Jacobians and relate them to the so-called genus-zero actions on
Riemann surfaces. We also give a complete classification and description of the
complex one-dimensional families of Riemann surfaces and Jacobians with a
generalized quaternion group action, extending known results concerning the
quasiplatonic case. Finally, we construct and describe explicit families of
abelian varieties with a quaternion group action and derive a period matrix for
the Jacobian of the surface with full automorphism group of second largest
order among the hyperelliptic surfaces of genus four.
