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.
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