$\newcommand{\Tors}{{\rm Tors}} \newcommand{\tf}{{\rm\, t.f.}} \newcommand{\Gt}{{\Gamma\!,\,\Tors}} \newcommand{\Gtf}{{\Gamma\!,\tf}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\HH}{{\mathbb H}}$Let $A$ be an abelian group. We denote by $A_\Tors$ the torsion subgroup of $A$. We set $A_\tf=A/A_\Tors\,$, which is a torsion free group.

Let $\Gamma$ be a finite group, and let $M$ be a $\Gamma$-module, that is, an abelian group on which $\Gamma$ acts. We denote by $M_\Gamma$ the group of coinvariants of $\Gamma$ in $M$, that is, $$ M_\Gamma=M\, \big/\bigg\{\sum_{\gamma\in \Gamma}(\,{}^\gamma y_\gamma-y_\gamma\,) \ \big|\ y_\gamma\in M\bigg\}.$$ We write $M_\Gt:= (M_\Gamma)_\Tors$ (which is the torsion subgroup of $M_\Gamma$), $\ M_\Gtf=M_\Gamma/ M_\Gt\ $ (which is a torsion free group). We consider the functors: \begin{align*} &(\Gamma,M)\,\rightsquigarrow\, M_\Gt\,,\\ &(\Gamma,M)\,\rightsquigarrow\, M_\Gtf\otimes(\Q/\Z)=M_\Gamma\otimes (\Q/\Z). \end{align*}

**Theorem.**
Let $\Gamma$ be a finite group, and let
$$0\to M_1\xrightarrow{i} M_2\xrightarrow{j} M_3\to 0$$
be a short exact sequence of $\Gamma$-modules.
Then there exists a natural homomorphism
$$\delta\colon (M_3)_\Gt\to (M_1)_\Gamma\otimes (\Q/\Z)$$
such that the following sequence is exact:
\begin{multline*}
(M_1)_\Gt \xrightarrow{i_*} (M_2)_\Gt \xrightarrow{j_*} (M_3)_\Gt\xrightarrow{\delta}\\
(M_1)_\Gamma\otimes(\Q/\Z)\xrightarrow{i_*} (M_2)_\Gamma\otimes(\Q/\Z)\xrightarrow{j_*} (M_3)_\Gamma\otimes(\Q/\Z)\to 0.
\end{multline*}

Question.Is this exact sequence known? If not, does this follow easily from some more general known exact sequence?

I do have a proof, I am asking for a reference!

**Special cases.**
$\DeclareMathOperator\Gal{Gal}$Let $\Gamma=\Gal(E/F)$ be the Galois group
of a finite Galois extension $E/F$ of nonarchimedean local fields.
Assume that our $\Gamma$-modules $M_i$ for $i=1,2,3$ are finitely generated and torsion-free, and for each $i=1,2,3$, let $T_i$ be the corresponding algebraic $F$-torus, which splits over $E$, with cocharacter group $M_i$.
We have a short exact sequence of $F$-tori
$$ 1\to T_1\to T_2\to T_3\to 1$$
and the corresponding Galois cohomology exact sequence
$$
H^1(F,T_1)\to H^1(F,T_2)\to H^1(F,T_3)\xrightarrow{\delta}
H^2(F,T_1)\to H^2(F,T_2)\to H^2(F,T_3)\to 1,
$$
where $H^n(F,T_i):=H^n(\Gal(F_s/F), M_i\otimes F_s^\times)$
is the cohomology of the *profinite* group $\Gal(F_s/F)$.
This is our sequence in this special case.

More generally, let again $\Gamma=\Gal(E/F)$ be the Galois group of a finite Galois extension $E/F$ of nonarchimedean local fields. Assume that our $\Gamma$-modules $M_i$ for $i=1,2,3$ are finitely generated, but now we do not assume that they are torsion-free. For each $i=1,2,3$, we choose a torsion-free resolution $$ 0\to M_i^{-1}\to M_i^0\to M_i\to 0.$$ We can choose these resolutions compatibly, so that we obtain a short exact sequence of complexes of torsion-free $\Gamma$-modules $$0\to (M_1^{-1}\to M_1^0)\to (M_2^{-1}\to M_2^0) \to (M_3^{-1}\to M_3^0)\to 0$$ and a short exact sequence of complexes of $F$-tori $$0\to (T_1^{-1}\to T_1^0)\to (T_2^{-1}\to T_2^0) \to (T_3^{-1}\to T_3^0)\to 0.$$ Write $T_i^\bullet$ for the complex of tori $(T_i^{-1}\to T_i^0)$. Then we know the hypercohomology groups $\HH^n(F,T_i^\bullet)$: $$ \HH^1(F,T_i^\bullet)\cong(M_i)_\Gt,\quad\ \HH^2(F,T_i^\bullet)\cong(M_i)_\Gamma\otimes(\Q/\Z);$$ see M. Borovoi, Abelian Galois cohomology of reductive groups, Memoirs AMS 132(626), 1998, Proposition 4.1. We have a hypercohomology exact sequence \begin{multline*} \HH^1(F,T_1^\bullet)\to \HH^1(F,T_2^\bullet)\to \HH^1(F,T_3^\bullet)\xrightarrow{\delta} \HH^2(F,T_1^\bullet)\to \HH^2(F,T_2^\bullet)\to \HH^2(F,T_3^\bullet)\to 1. \end{multline*} This is our sequence in this special case.