  Li and Praeger classified finite nonabelian simple groups, it has only one or
two fusion classes of any certain value. As a by-product, they classified
m-CI-groups, which is critical in the research of Cayley graphs. In the paper,
we will consider generalized Cayley graphs. This concept is proposed by Marusic
et al. In the paper, (local) m- GCI-group is defined, and we get many
properties and characterizations based on the generalized Cayley isomorphism,
which are the key measures for the classification of (local) m-GCI-group. And
above all, we will give a classification of local 2-GCI-groups and 2-GCI-groups
for finite nonabelian simple groups.
