A Dynamical Lifting Problem For Additive Polynomials

We introduce a dynamical analogue of the lifting problem for Galois covers of algebraic curves and find a negative solution for the collection of additive, separable polynomials over ${\bar{\mathbb{F}}}_p$. We also explicitly compute the dimension of the space of linear conjugacy classes in $M_{p^m}({\bar{\mathbb{F}}}_p)$ which contain an additive, separable polynomial.