  We give a new characterization of silting subcategories in the stable
category of a Frobenius extriangulated category, generalizing the result of Di
et al. (J. Algebra 525 (2019) 42-63) about the Auslander-Reiten type
correspondence for silting subcategories over triangulated categories. More
specifically, for any Frobenius extriangulated category $\mathcal{C}$, we
establish a bijective correspondence between silting subcategories of the
stable category $\underline{\mathcal{C}}$ and certain covariantly finite
subcategories of $\mathcal{C}$. As a consequence, a characterization of silting
subcategories in the stable category of a Frobenius exact category is given.
This result is applied to homotopy categories over abelian categories with
enough projectives, derived categories over Grothendieck categories with enough
projectives as well as to the stable category of Gorenstein projective modules
over a ring $R$.
