If X is a property or a class of groups, an automorphism ϕ of a group G is X-finitary if there is a normal subgroup N of G centralized by ϕ such that G/N is an X-group. Groups of such automorphisms for G a module over some ring have been very extensively studied over many years. However, for groups in general almost nothing seems to have been done. In 2009 V. V. Belyaev and D. A. Shved considered the general case for X the class of finite groups. Here we look further at the finite case but our main results concern the cases where X is either the class of polycyclic-by-finite groups or the class of Chernikov groups. The latter presents a new perspective on some work of Ya. D. Polovitskiĭ in the 1960s, which seems to have been at least partially overlooked in recent years. Our polycyclic cases present a different view of work of S. Franciosi, F. de Giovanni and M. J. Tomkinson from 1990. We describe the polycyclic cases in terms of matrix groups over the integers, and the Chernikov case in terms of matrix groups over the complex numbers.