By operating integration of energy over E_g to E_g-1 on energy dependent neutron diffusion equation, we can get definitions of some group constant like group absorption cross section, group scattering cross section weighed by flux.
For diffusion coefficient, it's weighed by gradient of flux, not flux itself. It is reasonable because the diffusion depends on gradient of flux, not flux.
It's interesting that we have flux dependence( or gradient of flux) , which we have not known yet, in the definition of group constants. This is called a recursive problem. To tackle this, we assume the spatial part and energy part of flux is separable. And then, we further assume that we know the energy dependence. For example, we assume a Maxwellian-Boltzmann distribution for thermal neutrons.
Since the difficulty of group parameter constants are already solved, we can easily write down the two group diffusion equation. What following is similar in one group. We will have two kind of problems, source problems and criticality problems.