  For a complex abelian variety $A$ with endomorphism ring isomorphic to the
maximal order in a quartic CM-field $K$, the Igusa invariants $j_1(A),
j_2(A),j_3(A)$ generate an abelian extension of the reflex field of $K$. In
this paper we give an explicit description of the Galois action of the class
group of this reflex field on $j_1(A),j_2(A),j_3(A)$. We give a geometric
description which can be expressed by maps between various Siegel modular
varieties. We can explicitly compute this action for ideals of small norm, and
this allows us to improve the CRT method for computing Igusa class polynomials.
Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby
implying that the `isogeny volcano' algorithm to compute endomorphism rings of
ordinary elliptic curves over finite fields does not have a straightforward
generalization to computing endomorphism rings of abelian surfaces over finite
fields.
