We study admissible transformations and Lie symmetries of various classes of Kolmogorov equations, of Fokker-Planck equations and of heat equations with potentials in the case of space dimension one. To carry out the group classifications of classes of Kolmogorov and Fokker-Planck equations with the help of the method of mappings between classes, we map these classes to the class of heat equations with potentials by families of point transformations parameterized by arbitrary elements of the corresponding initial classes. As a result, the group classifications of the above classes with respect to the associated equivalence groups are reduced to finding all solutions to heat equations with potentials listed in Lie’s canonical group classification that are inequivalent with respect to the essential point symmetry groups of these equations. For the related subclasses of equations with time-independent coefficients, we explicitly present complete lists of Lie-symmetry extensions that are inequivalent with respect to the associated equivalence groups.