Let $\mathcal {G}^{(\lambda )}$ be a group scheme that specializes to ${\mathbb {G}}_a$ when $\lambda =0$ and to ${\mathbb {G}}_m$ when $\lambda \in A^\times$, and let $\psi ^{(l)}:\mathcal {G}^{(\lambda )}\rightarrow \mathcal {G}^{(\lambda ^{p^l})}$ denote the $l$-th Kummer-Artin-Schreier type homomorphism determined by $\lambda$. We construct the Cartier dual of $\textrm{Ker}(\psi ^{(l)})$ over a $\mathbb {Z}/p^n\mathbb {Z}$-algebra.