The relation between ordinary automata, their fixed analogs of extensions, fixed analogs of generalized state-input extension with fixed mappings on input alphabet and characteristic semigroups are considered. The necessary and sufficient conditions for a fixed analog of an extension and generalized state-input extension with fixed mappings on input alphabets of the ordinary automaton to be linearly realizable over the field GF(p) are given. The equations describing fixed analogs of an extension and generalized state-input extension with fixed mappings on input alphabets are also stated.