Enumeration of semi-Latin squares

An (n \times n)/k semi-Latin square is an n\times n square in which nk letters are placed so that there are k letters in each row-column intersection and that each letter occurs once per row and once per column. It may be regarded as a family of nk permutations of n objects subject to certain restrictions. Squares of a given size fall into strong isomorphism classes (interchange of rows and columns not permitted), which are grouped into weak isomorphism classes (interchange of rows and columns permitted). We use group theory, graph theory, design theory and computing to find all isomorphism classes of (4 \times 4)/k semi-Latin squares for k=2, 3, 4.