On the injectivity of certain homomorphisms between extensions of ${\widehat{\mathcal{G}}}^{(\lambda )}$ by ${\widehat{\mathbb{G}}}_m$ over a ${\mathbb{Z}}_{(p)}$-algebra

Let ${\widehat{\mathcal{G}}}^{(\lambda )}$ be a formal group scheme which deforms ${\widehat{\mathbb{G}}}_a$ to ${\widehat{\mathbb{G}}}_m$. And let ${\psi }^{(l)}:{\widehat{\mathcal{G}}}^{(\lambda )}\to {\widehat{\mathcal{G}}}^{({\lambda }^{p^l})}$ be the $l$-th Frobenius-type homomorphism determined by $\lambda$. We show that the homomorphism $({\psi }^{(l)}{)}^{\ast }:H_0^2({\widehat{\mathcal{G}}}^{({\lambda }^{p^l})},{\widehat{\mathbb{G}}}_m)\to H_0^2({\widehat{\mathcal{G}}}^{(\lambda )},{\widehat{\mathbb{G}}}_m)$ induced by ${\psi }^{(l)}$ is injective over a ${\mathbb{Z}}_{(p)}$-algebra under a suitable restriction on $\lambda$. In this situation, the Cartier dual of $\mathrm{K}\mathrm{e}\mathrm{r}({\psi }^{(l)})$, which is a finite group scheme of order $p^l$, is described over a $\mathbb{Z}\mathrm{/}(p^n)$-algebra.