@srilakshmi Logging does not happen at the process group level. Processors logging is based on the processor class. So there is nothing in the log output produced by a processor within a process group that is going to tell you in which process group that particular processor belongs.
That being said, you may be able to prefix every processor's name within the same Process group with some string that identifies the process group. This processor name would generally be included in the the log output produced by the processor. Then you may be able to use logback filters (have not tried this myself) to filter log output based on these unique strings. https://logback.qos.ch/manual/filters.html
NiFi bulletins (bulletins are log output to the NiFi UI and have a rolling 5 minute life in the UI) however do include details about the parent Process Group in which the component generating the bulletin resides. You could build a dataflow in yoru NiFi to handle bulletin notification through the use of theSiteToSiteBulletinReportingTaskwhich is used to send bulletin to a destination remote import port on a target NiFi. A dataflow on the target NiFi could be built to parse the received bulletin records by the bulletinGroupName json path property so that all records from same PG are kept together. These 'like' records could then be written out to local filesystem based on the PG name, remote system, used to send email notifications, etc...
Example of what a Bulletin sent using the SiteToSiteBulletinReportingTask looks like: