Why do group II elements conduct? Looking at the periodic table, group II elements like magnesium are known to be metallic, and yet they have full outer shells. So this means they should have full (valence) bands.
Now, last time I checked a completely full band does not conduct.
In a similar way, group IV elements like silicon, germanium, and tin (which is practically a metal!) are actually semiconductors. But, diamond is an insulator.
I am an undergraduate of physics (and a newcomer to this site). I would love to understand the resolution to this problem. I have just started learning solid state physics so my knowledge is somewhat limited.
Why do group II elements conduct and group IV elements not act like semiconductors?

Edit:
Why do people keep using "Fermi energy" and "Fermi level" interchangeably?
 A: Your intuition is probably biased by  one-dimensional periodic potential examples of band structures, where it is impossible to have a band level crossing (it would imply the existence of more than 2 independent solutions of a second ordere ordinary differential equation).
In more than 1D things are different and energy levels corresponding to more than two k-points are possible. This is the case of second column elements and also explains the possibility of a range of behaviors for the elements of group IV (Carbon is an insulator in the normal pressure diamond lattice, while Lead is a metal).
A: Even when an isolated atom has a filled shell, the electron bands in a solid crystal may be partially filled.  The reason is that bands that originate from different atomic orbitals may actually overlap in energy.  This is shown on the left in the figure below.

When the bands overlap, the lowest-energy state has some electrons displaced from the top of the lower band to the bottom of the upper band, so that the Fermi energy in both bands is the same.  This leaves partially filled bands, enabling conduction to occur.
