The lattice of periods of the everywhere finite integrals on the algebraic curve of genus 2: η2=F(ξ), where F(ξ) is a polynomial of degree 6 (or 5) is studied by means of the parametrisation of the associated Kummer surface, in terms of theta functions, and where possible of elliptic functions, of two complex variables (u, v). According to the distribution in the ξ plane of the six roots of the equation F(ξ)=0 (counting ∞ as a root when this equation is only of degree 5), the lattice may have any one of a number of symmetry groups, expressible as groups of unitary, or of unitary and antiunitary, transformations on (u, v). These groups are enumerated, and studied in detail. As groups of orthogonal transformations in the real four dimensional space of the real and imaginary parts of (u, v) they are brought into the context of the classification of such groups in Du Val [3].