Let $E$ be a supersingular elliptic curve over a finite field of characteristic $p$, and $\mathbb{F}_q\supset \mathbb{F}_{p^2}$ be a finite field large enough such that all (absolute) endomorphisms of $E$ is defined over $\mathbb{F}_q$. We write $G$ for the absolute Galois group of $\mathbb{F}_q$. It is well known that the Frobenius automorphism $\varphi$ is a (topological) generator of $G$. Let us fix a prime $\ell\neq p$. By the Tate Conjecture, $$\mathrm{End}(E)\otimes\mathbb{Q}_\ell=\mathrm{End}_{\mathbb{Q}_\ell}(V_\ell(E))^G=\mathrm{End}_{\mathbb{Q}_\ell}(V_\ell(E))^\varphi. $$

Since $E$ is supersingular, $\mathrm{End}(E)$ is an order in a quaternion algebra. In particular, $\mathrm{rank}_\mathbb{Z}(\mathrm{End}(E))=4$. So $$ \mathrm{End}(E)\otimes\mathbb{Q}_\ell=\mathrm{End}_{\mathbb{Q}_\ell}(V_\ell(E))$$ It follows that $\varphi$ is in the center of $\mathrm{End}_{\mathbb{Q}_\ell}(V_\ell(E))$. In other words, $\varphi$ is a scalar. This clearly leads to a contradiction (say, with the Riemann Hypothesis). Where did the argument go wrong?