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Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated free $\mathbb{Z}[G]$-modules), with kernel $K$. For every $n\geq 1$, let $M_n:=\phi^{-1}(nN)$. Since $nN\cong N$, we have an exact sequence $0\to K\to M_n \to N\to 0$. I was wondering if it is possible to determine the class of this extension in terms of $n$ and the class of $0\to K\to M\to N\to 0$ inside the finite abelian group $Ext^1_G(N,K)$?

For example, is it true that $[M_n]=a+bn$, for some $a,b\in Ext^1_G(N,K)$?

Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated $\mathbb{Z}[G]$-modules, free as $\mathbb Z$-modules), with kernel $K$. For every $n\geq 1$, let $M_n:=\phi^{-1}(nN)$. Since $nN\cong N$, we have an exact sequence $0\to K\to M_n \to N\to 0$. I was wondering if it is possible to determine the class of this extension in terms of $n$ and the class of $0\to K\to M\to N\to 0$ inside the finite abelian group $Ext^1_G(N,K)$?

For example, is it true that $[M_n]=a+bn$, for some $a,b\in Ext^1_G(N,K)$?

Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated free $\mathbb{Z}[G]$-modules), with kernel $K$. For every $n\geq 1$, let $M_n:=\phi^{-1}(nN)$. Since $nN\cong N$, we have an exact sequence $0\to K\to M_n \to N\to 0$. I was wondering if it is possible to determine the class of this extension in terms of $n$ and the class of $0\to K\to M\to N\to 0$ inside the finite abelian group $Ext^1_G(N,K)$?

Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated free $\mathbb{Z}[G]$-modules), with kernel $K$. For every $n\geq 1$, let $M_n:=\phi^{-1}(nN)$. Since $nN\cong N$, we have an exact sequence $0\to K\to M_n \to N\to 0$. I was wondering if it is possible to determine the class of this extension in terms of $n$ and the class of $0\to K\to M\to N\to 0$ inside the finite abelian group $Ext^1_G(N,K)$?

For example, is it true that $[M_n]=a+bn$, for some $a,b\in Ext^1_G(N,K)$?