$G/G_n$. Choose a finite presentation of $M$: $$kG^r\xrightarrow{f}kG^s\to M\to 0.$$ Let $f_n=k[G/G_n]\otimes_{kG}f$. By right-exactness of tensor products we have the exact sequence $$k[G/G_n]^r\xrightarrow{f_n}k[G/G_n]^s\to k[G/G_n]\otimes_{kG} M\to
  0.$$ The natural map $Q_n\subset G\rightarrow G/G_n$ induces an isomorphism $j_n\colon k[Q_n]\to k[G/G_n]$ of $k$-vector spaces. The map $f$ is given by right multiplication $f=R_A$ with a matrix $A\in
  M_{r,s}(kG)$. Viewing $A$ as a map $G\to k^{r\times s}$ let ${\operatorname{supp}}(A)$ be the support of $A$. Let $R > 0$ be the diameter of ${\operatorname{supp}}(A)\cup{\operatorname{supp}}(A)^{-1}$ (with respect to the fixed word metric on $G$). Then $f$ restricts to a map $$f\vert_{Q_n\setminus \partial_RQ_n}\colon
  k[Q_n\setminus \partial_RQ_n]^r\to k[Q_n]^s.$$ Hence there is precisely one $k$-linear map $g$ for which the following diagram of $k$-vector spaces commutes: $$\label{eq:approximation_diagram}
  \xymatrix@C=16mm{k[G/G_n]^r\ar[r]^{f_n}& k[G/G_n]^s\ar[r]&k[G/G_n]\otimes_{kG}M\ar[r]&0\\
    k[Q_n\setminus \partial_RQ_n]^r\ar[u]^{j_n\vert_{Q_n\setminus \partial_RQ_n}}
   \ar[r]^-{f\vert_{Q_n\setminus \partial_RQ_n}} &
    k[Q_n]^s\ar[u]^{j_n}_{\cong} \ar[r]^{{\operatorname{pr}}}&{\operatorname{coker}}(
    f\vert_{Q_n\setminus \partial_RQ_n})\ar@{->>}[u]^g\ar[r]&0}$$ One easily verifies that $g$ is surjective and that $$\ker(g)\subset{\operatorname{im}}\bigl({\operatorname{pr}}\circ j_n^{-1}\circ f_n\colon
  k[G/G_n]^r\to {\operatorname{coker}}( f\vert_{Q_n\setminus \partial_RQ_n})\bigr).$$ The map ${\operatorname{pr}}\circ j_n^{-1}\circ f_n$ descends to a map $${\operatorname{pr}}\circ j_n^{-1}\circ f_n\colon
  {\operatorname{coker}}(j_n\vert_{Q_n\setminus \partial_RQ_n})\to {\operatorname{coker}}(
  f\vert_{Q_n\setminus \partial_RQ_n}).$$ Note that $$\dim_k\bigl({\operatorname{coker}}(j_n\vert_{Q_n\setminus \partial_RQ_n})\bigr)=
  r \cdot |\partial_R Q_n|.$$ Thus, $$\dim_k({\operatorname{coker}}(  f\vert_{Q_n\setminus \partial_RQ_n}))-\dim_k\left(k[G/G_n]\otimes_{kG}M\right)
    =\dim_k \ker(g)\le r\cdot \bigl|\partial_R Q_n\bigr|.$$ By replacing the upper row in diagram  by $$k[Q_n]^r\xrightarrow{{\operatorname{pr}}[Q_n]\circ f\circ j[Q_n]}k[Q_n]^s\to M[Q_n]\to 0$$ and essentially running the same argument as before 