The Connecticut General Statutes prescribe a rather strict timeline for the pistol permit process. This timeline is dictated by CGS29-28a and CGS29-29. The deadlines described in these statutes give your local issuing authority "5 days" from the date you submit your application with fingerprints until they must submit your fingerprints to the State Police Bureau of Identification (SPBI). Once your FBI report is received back from the SPBI, the local issuing authority has "one week" to issue an approval or denial.
The entire process is required to take no more than 8 weeks by statute.