Frobenius functors and Gorenstein projective precovers

We establish relations between Gorenstein projective precovers linked by Frobenius functors. This is motivated by an open problem that how to find general classes of rings for which modules have Gorenstein projective precovers. It is shown that if $F:\C\rightarrow\D$ is a separable Frobenius functor between abelian categories with enough projective objects, then every object in $\C$ has a Gorenstein projective precover provided that every object in $\D$ has a Gorenstein projective precover. This result is applied to separable Frobenius extensions and excellent extensions.