  It is known that average Siegel theta series lie in the space of Siegel
Eisenstein series. Also, every lattice equipped with an even integral quadratic
form lies in a maximal lattice. Here we consider average Siegel theta series of
degree 2 attached to maximal lattices; we construct maps for which the average
theta series is an eigenform. We evaluate the action of these maps on Siegel
Eisenstein series of degree 2 (without knowing their Fourier coefficients), and
then realise the average theta series as an explicit linear combination of the
Eisenstein series.
