  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.
