  We describe the supercharacter theories of the semidirect product of H and K,
$H\rtimes K$ in terms of the supercharacter theories of the direct product of H
and K in the case when both H and K are Abelian groups. To do this we introduce
the concept of a homomorphism of supercharacter theories. This provides a
classification of the supercharacter theories of the dihedral groups of order
2m when m is odd using the known classification of the supercharacter theories
of cyclic groups.
