A hypersubstitution maps an algebra to an algebra of the same type, by replacing the operations by term operations. A hypersubstitution is called proper with respect to a variety if it is a mapping on this variety and it is called inner if it is an identity mapping on this variety. Proper as well as inner hypersubstitutions characterize a variety. Each inner hypersubstitution is a proper one but not conversely. In the present paper, we characterize the relationship between proper and inner hypersubstitutions for varieties of rings satisfying $x^{n+1}\approx x$, in particular for $n=6$.