Edit. The OP has clarified that all of the sheaves are sheaves of locally constant functions, rather than sheaves of continuous functions or $C^\infty$ functions. In that case, it is better to consider principal bundles that are induced from an Abelian subgroup of the normalizer of the maximal torus that is not contained in the maximal torus. For instance, let $\Gamma'$ be the subgroup of $\textbf{PGL}(n,\mathbb{C})$ generated by the equivalence classes of the following two matrices, $$ A=\left[\begin{array}{cccccc} 0 & 1 & 0 & \dots & 0 & 0\\ 0 & 0 & 1 & \dots & 0 & 0\\ 0 & 0 & 0 & \dots & 0 &0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 &\dots & 0 & 1 \\ 1 & 0 & 0 & \dots & 0 & 0 \end{array} \right], \ \ B=\left[\begin{array}{cccccc} 1 & 0 & 0 & \dots & 0 & 0\\ 0 & z & 0 & \dots & 0 & 0 \\ 0 & 0 & z^2 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 &\dots & z^{n-2} & 0 \\ 1 & 0 & 0 & \dots & 0 & z^{n-1} \end{array} \right], $$ where $z$ is a primitive $n^{\text{th}}$ root of unity. The group $\Gamma'$ is isomorphic to $(\mathbb{Z}/n\mathbb{Z})\times \mu_n$. The inverse image $\Gamma$ of $\Gamma'$ in $\textbf{SL}(n,\mathbb{C})$ is a non-Abelian group. It is a central extension of $\Gamma$ by $\mu_n\cdot \text{Id}_{n\times n}$. The commutation relation is $ABA^{-1}B^{-1} = z\cdot \text{Id}_{n\times n}$. The central extension of finite groups, $$ 1 \to \mu_n\cdot \text{Id}_{n\times n} \to \Gamma \to (\mathbb{Z}/n\mathbb{Z})\times \mu_n \to 1,$$ gives a long exact sequence of non-Abelian cohomology with a connecting map, $$\delta:H^1(M;(\mathbb{Z}/n\mathbb{Z})\times \mu_n) \to H^2(M;\mu_n).$$ This connecting map is what is usually called the Weil pairing. There are many ways to construct the Weil pairing; the construction above is just one (probably not the best, since it is a little unclear how this behaves when you replace the integer $n$ by a multiple of $n$).

Edit. The OP has clarified that all of the sheaves are sheaves of locally constant functions, rather than sheaves of continuous functions or $C^\infty$ functions. In that case, it is better to consider principal bundles that are induced from an Abelian subgroup of the normalizer of the maximal torus that is not contained in the maximal torus. For instance, let $\Gamma'$ be the subgroup of $\textbf{PGL}(n,\mathbb{C})$ generated by the equivalence classes of the following two matrices, $$ A=\left[\begin{array}{cccccc} 0 & 1 & 0 & \dots & 0 & 0\\ 0 & 0 & 1 & \dots & 0 & 0\\ 0 & 0 & 0 & \dots & 0 &0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 &\dots & 0 & 1 \\ 1 & 0 & 0 & \dots & 0 & 0 \end{array} \right], \ \ B=\left[\begin{array}{cccccc} 1 & 0 & 0 & \dots & 0 & 0\\ 0 & z & 0 & \dots & 0 & 0 \\ 0 & 0 & z^2 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 &\dots & z^{n-2} & 0 \\ 1 & 0 & 0 & \dots & 0 & z^{n-1} \end{array} \right], $$ where $z$ is a primitive $n^{\text{th}}$ root of unity. The group $\Gamma'$ is isomorphic to $(\mathbb{Z}/n\mathbb{Z})\times \mu_n$. The inverse image $\Gamma$ of $\Gamma'$ in $\textbf{SL}(n,\mathbb{C})$ is a non-Abelian group. It is a central extension of $\Gamma'$ by $\mu_n\cdot \text{Id}_{n\times n}$. The commutation relation is $ABA^{-1}B^{-1} = z\cdot \text{Id}_{n\times n}$. The central extension of finite groups, $$ 1 \to \mu_n\cdot \text{Id}_{n\times n} \to \Gamma \to (\mathbb{Z}/n\mathbb{Z})\times \mu_n \to 1,$$ gives a long exact sequence of non-Abelian cohomology with a connecting map, $$\delta:H^1(M;(\mathbb{Z}/n\mathbb{Z})\times \mu_n) \to H^2(M;\mu_n).$$ This connecting map is what is usually called the Weil pairing. There are many ways to construct the Weil pairing; the construction above is just one (probably not the best, since it is a little unclear how this behaves when you replace the integer $n$ by a multiple of $n$).

I will add some details. Let $M$ be a compact, closed, oriented, real manifold of real dimension $2$. Denote by $\textbf{SL}(n,\mathbb{C})$, resp. $\textbf{PGL}(n,\mathbb{C})$, the real Lie group of $n\times n$ complex matrices with determinant $1$, resp. equivalence classes of $n\times n$ complex matrices up to scaling.

Edit. The OP has clarified that all of the sheaves are sheaves of locally constant functions, rather than sheaves of continuous functions or $C^\infty$ functions. In that case, it is better to consider principal bundles that are induced from an Abelian subgroup of the normalizer of the maximal torus that is not contained in the maximal torus. For instance, let $\Gamma'$ be the subgroup of $\textbf{PGL}(n,\mathbb{C})$ generated by the equivalence classes of the following two matrices, $$ A=\left[\begin{array}{cccccc} 0 & 1 & 0 & \dots & 0 & 0\\ 0 & 0 & 1 & \dots & 0 & 0\\ 0 & 0 & 0 & \dots & 0 &0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 &\dots & 0 & 1 \\ 1 & 0 & 0 & \dots & 0 & 0 \end{array} \right], \ \ B=\left[\begin{array}{cccccc} 1 & 0 & 0 & \dots & 0 & 0\\ 0 & z & 0 & \dots & 0 & 0 \\ 0 & 0 & z^2 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 &\dots & z^{n-2} & 0 \\ 1 & 0 & 0 & \dots & 0 & z^{n-1} \end{array} \right], $$ where $z$ is a primitive $n^{\text{th}}$ root of unity. The group $\Gamma'$ is isomorphic to $(\mathbb{Z}/n\mathbb{Z})\times \mu_n$. The inverse image $\Gamma$ of $\Gamma'$ in $\textbf{SL}(n,\mathbb{C})$ is a non-Abelian group. It is a central extension of $\Gamma$ by $\mu_n\cdot \text{Id}_{n\times n}$. The commutation relation is $ABA^{-1}B^{-1} = z\cdot \text{Id}_{n\times n}$. The central extension of finite groups, $$ 1 \to \mu_n\cdot \text{Id}_{n\times n} \to \Gamma \to (\mathbb{Z}/n\mathbb{Z})\times \mu_n \to 1,$$ gives a long exact sequence of non-Abelian cohomology with a connecting map, $$\delta:H^1(M;(\mathbb{Z}/n\mathbb{Z})\times \mu_n) \to H^2(M;\mu_n).$$ This connecting map is what is usually called the Weil pairing. There are many ways to construct the Weil pairing; the construction above is just one (probably not the best, since it is a little unclear how this behaves when you replace the integer $n$ by a multiple of $n$).

Because $\Gamma$ is a nilpotent group of nilpotency class $2$, every group homomorphism, $$\rho:\pi_1(M;\ast) \to \Gamma,$$ factors through the $2$-nilpotent quotient $\Pi(M)$ of $\pi_1(M;\ast)$. This fits into a central extension, $$0 \to \left(\bigwedge^2_{\mathbb{Z}} H_1(M;\mathbb{Z})\right)/H_2(M;\mathbb{Z})\to \Pi \to H_1(M;\mathbb{Z})\to 0,$$ where a generator $\gamma$ of $H_2(M;\mathbb{Z})$ maps to the usual relation in $\bigwedge^2H_1(M;\mathbb{Z})$, $$\gamma \mapsto \alpha_1\wedge \beta_1 + \dots + \alpha_g\wedge \beta_g$$ for the standard presentation, $$\pi_1(M,\ast) = \langle \alpha_1,\dots,\alpha_g,\beta_1,\dots,\beta_g: [\alpha_1,\beta_1]\cdots [\alpha_g,\beta_g] \rangle.$$

So now, for every integer $m$, consider the homomorphism $$\sigma_m: \Pi \to \Gamma',$$ $$\alpha_1 \mapsto [A], \ \beta_1 \mapsto [B^m], \ \alpha_{i>1} \mapsto [\text{Id}_{n\times n}], \ \beta_{j>1}\mapsto [\text{Id}_{n\times n}].$$ This maps $\gamma$ to $z^m\cdot \text{Id}_{n\times n}$. Thus, $\sigma_m$ lifts to a representation $\rho_m:\Pi \to \Gamma$ if and only if $m$ is divisible by $n$. When $m$ is not divisible by $m$, the commutator of $A$ and $B^m$ maps to $z^m\text{Id}_{n\times n}$. Thus the corresponding $\Gamma'$-principal bundle over $M$ corresponding to $\sigma_m$ has image under $\delta$ equal to $m$ times the generator of $H^2(M;\mu_n)$.

Since the long exact sequence in non-Abelian cohomology associated to a central extension is functorial, it follows that for the associated $\textbf{PGL}(n,\mathbb{C})$-principal bundle over $M$, the image under $\delta$ also equals $m$ times the generator. Thus, $\delta$ is surjective.

Original post. I will add some details. Let $M$ be a compact, closed, oriented, real manifold of real dimension $2$. Denote by $\textbf{SL}(n,\mathbb{C})$, resp. $\textbf{PGL}(n,\mathbb{C})$, the real Lie group of $n\times n$ complex matrices with determinant $1$, resp. equivalence classes of $n\times n$ complex matrices up to scaling.