Any group is represented by an outerautomorphism group

Recently S. Kojima showed that any finite group is isomorphic to the outerautomorphism (class) group Out(;)=Aut(;)/Inn(;) of some discrete subgroup ; of the projective special linear group PSL(2,C). The paper generalizes this result and proved that any group G is isomorphic to the outerautomorphism group Out(;) of some group ;. The method of constructing ; depends on the graph of groups associated to a modified Cayley graph of G. The edge and vertex groups are carefully chosen among the Mathieu groups and their subgroups.