On the injectivity of certain homomorphisms between extensions of G(λ) by Gₘ over a Z₍ₚ₎-algebra

Let G(λ) be a formal group scheme which deforms Gₐ to Gₘ. And let ψ⁽ˡ⁾:G(λ)→G^(λpˡ) be the l-th Frobenius-type homomorphism determined by λ. We show that the homomorphism (ψ⁽ˡ⁾)^:H²₀(G^(λpˡ),Gₘ)→ H²₀(G(λ),Gₘ) induced by ψ⁽ˡ⁾ is injective over a Z₍ₚ₎-algebra under a suitable restriction on λ. In this situation, the Cartier dual of Ker(ψ⁽ˡ⁾), which is a finite group scheme of order pˡ, is described over a Z/(pⁿ)-algebra.