On Nielsen type invariant for varieties over a finite field

For a self-map of finite CW complex X one has a homotopy invariant: for each positive integer n one gets a finite linear combination with integer coefficients of conjugacy classes of the fundamental group of the mapping cylinder of period n with respect to the base circle. Algebraically, this sequence can be described in terms of a finite super-matrix with coefficients in the group ring of the fundamental group, by taking traces of powers of this matrix. 

For varieties over finite field one also gets such Nielsen type invariants, by considering formal sums over points over finite extensions of corresponding Frobenius automorphisms.

I propose a conjecture that these sequences of sums are again described by matrices in group rings, as in the topological case.