An equivariant smash spectral sequence and an unstable box product

Let $G$ be a finite group. We construct a first quadrant spectral sequence which converges to the equivariant homotopy groups of the smash product $X \wedge Y$ for suitably connected, based $G$-CW complexes $X$ and $Y$. The $E^2$ term is described in terms of a tensor product functor of equivariant $\Pi$-algebras. A homotopy version of the non-equivariant Künneth theorem and the equivariant suspension theorem of Lewis are both shown to be special cases of the corner of the spectral sequence. We also give a categorical description of this tensor product functor which is analogous to the description in equivariant stable homotopy theory of the box product of Mackey functors. For this reason, the tensor product functor deserves to be called an ``unstable box product''.