On c-homogeneous ordered sets

An ordered set (X, ⩽) is called c-homogeneous if every automorphism of a connected induced suborder of (X, ⩽) can be extended to an automorphism of (X, ⩽). We show that a finite connected ordered set (X, ⩽) is c-homogeneous iff (X, ⩽) is a crown, a standard n-dimensional ordered set, or a weak order.