jmc

The base of the pyramid is a square of side length $2$, and thus has area $2^2=4$. The height of the pyramid is half the height of the cube, or $\frac{1}{2}\cdot 2=1$. Therefore, the volume of the pyramid is $$\begin{array}{rl}\frac{1}{3}\cdot (\text{area of base})\cdot (\text{height})& =\frac{1}{3}\cdot 4\cdot 1\\ & =\boxed{\frac{4}{3}}.\end{array}$$