The etale -sheaves on have a natural isomorphism . The definition of Weil sheaves captures this feature, see page 7, section 1.1 def. 1.2 of Weil Conjectures, Perverse Sheaves & ℓ-adic Fourier Transform. The idea is that the Weil group acts on a Weil sheaf . More precisely, there is an equivalence between the category of lisse Weil sheaves and the category of continuous representations of the Weil group on finite dimensional vector spaces over . The motivation is Galois descent theory tells us that there is an equivalence of categories between constructible -sheaves on and constructible -sheaves on with a specified action of . But if we are in a setting where we know that such an isomorphism exists, but we cannot immediately verify that it is continuous, we are led to the definition of Weil sheaves.
The definition of Weil sheaves use geometric Frobenius, which induces a Frobenius automorphism (over not ), but the actions of (relative Frobenius) and the Galois action of the Frobenius induced both on geometric points and on cohomology with compact support coincide so it doesn’t matter.
An important theorem for Weil sheaves is that after twisting, they are essentially the same as etale -sheaves. First, by Proposition 1.1.12 of the note, a -representation of on extends to a representation of iff some (equivalently, any) degree-1 element acts with eigenvalues which are -adic units. We want to use this theorem to prove the following:
Suppose that is normal and geometrically connected. Then, an irreducible lisse Weil sheaf of rank is an actual -sheaf if and only if its determinant is.
Note that the associated representation of is at least geometrically semisimple, i.e. its restriction to the geometric fundamental group is semisimple since it is a continuous representation of a compact group. We can then apply Theorem 1.3.3 of the book to get that Zariski closure of is a semisimple algebraic group. The key theorem is Theorem 1.3.1, which says that the image of the geometric fundamental group under a continuous character of the Weil group is a finite group, whose proof uses geometric class field theory. See this note for the detail. This also implies for some , we can write for some and . Since is semisimple, by increasing we can assume . Once we prove this, the rest boils down to proving all eigenvalues of are -adic units. This is essentially a compactness argument, using the adjoint action of on (Note that stablizes the image of the geometric fundamental group because and does.)
Reference:
https://math.stanford.edu/~conrad/Weil2seminar/Notes/L19.pdf
https://math.stanford.edu/~conrad/Weil2seminar/Notes/L22.pdf