Let $ E$ be a supersingular elliptic curve over $ \mathbf{F}_p$ , and $ H$ its endomorphism algebra $ \text{End}(E)\otimes_{\mathbf{Z}}\mathbf{Q}$ , a quaternion algebra (non split at $ p$ and $ \infty$ ).
For every prime $ \ell\neq p$ , there is a faithful $ \ell$ adic algebrarepresentation:
$ $ \rho_{\ell} : H\to \text{End}_{\mathbf{Q}_{\ell}}(V_{\ell}(E))$ $
where $ V_{\ell}(E) := (\varprojlim_{n\ge 0} E_{\overline{\mathbf{F}}_p}({\overline{\mathbf{F}}_p})[\ell^n])\otimes_{\mathbf{Z}_{\ell}}\mathbf{Q}_{\ell}$ is the rational $ \ell$ adic Tate module of $ E$ .

Using the natural isomorphism $ H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell}) \simeq \text{Hom}_{\mathbf{Q}_{\ell}}(V_{\ell}(E),\mathbf{Q}_{\ell})$ we have an $ H\otimes_{\mathbf{Q}}\mathbf{Q}_{\ell}$ module structure on $ H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ , as is classically known.

Now we make a more naive construction. Functoriality of the étale site of $ E_{\overline{\mathbf{F}}_p}$ gives that for any endomorphism $ f : E\to E$ there is a map $ $ H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})\to H^i(E_{\overline{\mathbf{F}}_p},f^{1}\mathbf{Q}_{\ell})$ $ and since $ \mathbf{Q}_{\ell}$ is constant (here I am being imprecise about the nature of $ \mathbf{Q}_{\ell}$ , which is not a constant sheaf on the étale site, but the meaning is clear from the context) we also have an isomorphism $ H^i(E_{\overline{\mathbf{F}}_p},f^{1}\mathbf{Q}_{\ell})\simeq H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ , and we call $ $ f^* : H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})\to H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ $ the composition of the two. In other words, every element $ f\in\text{End}(E)$ has an effect $ f^*$ on $ H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ . On the other hand, the effect of each element of $ \mathbf{Z}\subset\text{End}(E)$ on $ H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ is invertible, and so the above construction defines, for every element $ f\in H$ , an effect $ f^*$ on $ \ell$ adic cohomology.
Does the construction in the second point, for $ i=1$ , give an action of $ H$ on $ H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ as an algebra? If so, does this action agree with the one constructed in the first point?
My expectation is that the answer to the first, and hence second, question is no.
The second construction should only define on $ H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ the structure of representation of $ H^{\times}$ , and it should not be possible to upgrade this to an algebra action of $ H$ .
Existence of the $ \ell$ adic Tatemodule functor, and the fact that for abelian varieties $ A,B$ the map $ \text{Hom}_{\rm AV}(A,B)\otimes_{\mathbf{Z}}\mathbf{Q}_{\ell}\to \text{Hom}_{\mathbf{Q}_{\ell}}(V_{\ell}(A),V_{\ell}(B))$ is injective, should be crucial to have an algebra action of $ H$ on $ H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ , and it feels it should not be possible to construct it just as a consequence of functoriality of the étale site, a much less deep fact.