Let R be a commutative Noetherian ring and I a proper ideal of R. In this paper, we study finitely generated R-modules M with only one non-vanishing local cohomology module H I c (M) where c = c d (I , M) . Let C be a semidualizing R-module. We investigate the conditions under which H I c (M) belongs to either the Auslander class A C (R) or the Bass class B C (R) .