On cardinalities in quotients of inverse limits of groups

Let λ be א0 or a strong limit of cofinality א0. Suppose that 〈Gm, πm,n : m ≤ n < ω〉 and 〈Hm, π t m,n : m ≤ n < ω〉 are projective systems of groups of cardinality less than λ and suppose that for every n < ω there is a homorphism σ : Hn → Gn such that all the diagrams commute. If for every μ < λ there exists 〈fi ∈ Gω : i < μ〉 such that i 6= j =⇒ fif −1 j 6∈ σω(Hω) then there exists 〈fi ∈ Gω : i < 2 〉 such that i 6= j =⇒ fif −1 j 6∈ σω(Hω).