Let $k$ be a field and $\bar k$ a separable closure. For an algebraic $k$-torus, denote by $G_* = \mathrm{Hom}(\mathbb{G}_{m, \bar k}, G_{\bar k})$ its group of cocharacters. This is a finitely generated, free abelian group with a continuous $\mathrm{Gal}_k = \mathrm{Gal}(\bar k , k)$ action. Then it is claimed at various places that there exists a natural isomorphism of $\mathrm{Gal}_{k}$-modules $$ G_* \otimes_{\mathbb{Z}} \mathbb{Z}_{\ell}(1) \cong T_\ell(G) $$ where $T_\ell(G)$ denotes the $\ell$-adic Tate module of $G$, i.e. $$ T_\ell(G) = \varprojlim_{n} G[\ell^n](\bar k) $$ Here $G[\ell^n](\bar k)$ denotes $\bar k$-valued $\ell^n$-torsion points of $G$.
Since $G_*$ is a flat and finitely-presented $\mathbb{Z}$-module, it suffices to show $$ G_* \otimes \mu_{\ell^n} \cong G[\ell^n](\bar k) $$ I am unable to find a proof of this nor did I find a reference. Any suggestion is appreciated.
There is an isomorphism $$\begin{align*} \mathrm{Hom}(\mathbb G_{m,\overline k},G_{\overline k})\otimes \mu_{\ell^n}&\simeq G[\ell^n](\overline k)\\ \varphi\otimes z&\mapsto \varphi(z). \end{align*}$$