We consider a clustered network with small-world subnetworks of inhibitory fast spiking interneurons and investigate the effect of intermodular connection on the emergence of fast sparsely synchronized rhythms by varying both the intermodular coupling strength J(inter) and the average number of intermodular links per interneuron M(syn)(inter). In contrast to the case of nonclustered networks, two kinds of sparsely synchronized states such as modular and global synchronization are found. For the case of modular sparse synchronization, the population behavior reveals the modular structure, because the intramodular dynamics of subnetworks make some mismatching. On the other hand, in the case of global sparse synchronization, the population behavior is globally identical, independently of the cluster structure, because the intramodular dynamics of subnetworks make perfect matching. We introduce a realistic cross-correlation modularity measure, representing the matching degree between the instantaneous subpopulation spike rates of the subnetworks, and examine whether the sparse synchronization is global or modular. Depending on its magnitude, the intermodular coupling strength J(inter) seems to play "dual" roles for the pacing between spikes in each subnetwork. For large J(inter), due to strong inhibition it plays a destructive role to "spoil" the pacing between spikes, while for small J(inter) it plays a constructive role to "favor" the pacing between spikes. Through competition between the constructive and the destructive roles of J(inter), there exists an intermediate optimal J(inter) at which the pacing degree between spikes becomes maximal. In contrast, the average number of intermodular links per interneuron M(syn)(inter) seems to play a role just to favor the pacing between spikes. With increasing M(syn)(inter), the pacing degree between spikes increases monotonically thanks to the increase in the degree of effectiveness of global communication between spikes. Furthermore, we employ the realistic sub- and whole-population order parameters, based on the instantaneous sub- and whole-population spike rates, to determine the threshold values for the synchronization-unsynchronization transition in the sub- and whole populations, and the degrees of global and modular sparse synchronization are also measured in terms of the realistic sub- and whole-population statistical-mechanical spiking measures defined by considering both the occupation and the pacing degrees of spikes. It is expected that our results could have implications for the role of the brain plasticity in some functional behaviors associated with population synchronization.