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.