  We study sources of isomorphisms of additive cellular automata on finite
groups (called index-group). It is shown that many isomorphisms (called
regular) of automata are reducible to the isomorphisms of underlying algebraic
structures (such as the index-group, monoid of automata rules, and its subgroup
of reversible elements). However for some groups there exist not regular
automata isomorphisms. A complete description of linear automorphisms of the
monoid is obtained. These automorphisms cover the most part of all automata
isomorphisms for small groups and are represented by reversible matrices M such
that for any index-group circulant C the matrix M^{-1}CM is an index-group
circulant.
