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

Let $\widehat{\mathcal{G}}^{(λ)}$ be a formal group scheme which deforms $\widehat{\mathbb{G}}_a$ to $\widehat{\mathbb{G}}_m$. And let $ψ^{(l)}:\widehat{\mathcal{G}}^{(λ)}\rightarrow\widehat{\mathcal{G}}^{(λ^{p^l})}$ be the $l$-th Frobenius-type homomorphism determined by $λ$. We show that the homomorphism $(ψ^{(l)})^\ast:H^2_0(\widehat{\mathcal{G}}^{(λ^{p^l})},\widehat{\mathbb{G}}_m)\rightarrow H^2_0(\widehat{\mathcal{G}}^{(λ)},\widehat{\mathbb{G}}_m)$ induced by $ψ^{(l)}$ is injective over a $\mathbb{Z}_{(p)}$-algebra under a suitable restriction on $λ$. In this situation, the Cartier dual of $\mathrm{Ker}(ψ^{(l)})$, which is a finite group scheme of order $p^l$, is described over a $\mathbb{Z}/(p^n)$-algebra.