The Cayley isomorphism property for Cayley maps

In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~ref{111015a} where a short list of possible candidates for CIM-groups is given. Theorem~ref{111015c} provides concrete examples of infinite series of CIM-groups.