  We evaluate the action of Hecke operators on Siegel Eisenstein series of
degree 2, square-free level and arbitrary character, without using knowledge of
their Fourier coefficients. From this we construct a basis of simultaneous
eigenforms for the full Hecke algebra, and we compute their eigenvalues. As
well, we obtain Hecke relations among the Eisenstein series. Using these Hecke
relations in the case that $\stufe$ is square-free and the character is
trivial, we generate a basis for the space of Eisenstein series.
