  Let $\widehat{\mathcal{G}}^{(\lambda)}$ be a formal group scheme which
deforms $\widehat{\mathbb{G}}_a$ to $\widehat{\mathbb{G}}_m$. And let
$\psi^{(l)}:\widehat{\mathcal{G}}^{(\lambda)}\rightarrow\widehat{\mathcal{G}}^{(\lambda^{p^l})}$
be the $l$-th Frobenius-type homomorphism determined by $\lambda$. We show that
the homomorphism
$(\psi^{(l)})^\ast:H^2_0(\widehat{\mathcal{G}}^{(\lambda^{p^l})},\widehat{\mathbb{G}}_m)\rightarrow
H^2_0(\widehat{\mathcal{G}}^{(\lambda)},\widehat{\mathbb{G}}_m)$ 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
$\mathrm{Ker}(\psi^{(l)})$, which is a finite group scheme of order $p^l$, is
described over a $\mathbb{Z}/(p^n)$-algebra.
