We consider the integer group determinants for groups that are semidirect products of $\mathbb Z_p$ and $\mathbb Z_n$ with $p$ prime and $n\mid p-1$. We give a complete description of the integer group determinants for the general affine groups of degree one GA(1,5) and GA(1,7), and for $\mathbb Z_7\rtimes \mathbb Z_3,$ $\mathbb Z_{11}\rtimes \mathbb Z_5$ and $\mathbb Z_{13}\rtimes \mathbb Z_6,$ showing that the obvious divisibility and congruence conditions arising from the form of the group determinant when $n=p-1$ or $\frac{1}{2}(p-1)$, can be sufficient as well as necessary for these types of groups.
