On Windows, the group is created as a local Windows group into which you need to add your user account (or group). The local group name is not compiled into the Oracle binaries.
In UNIX, it's possible when using SQL*Plus to connect to an idle instance AS SYSDBA, you get “ORA-01031: insufficient privileges” even though you have changed the $ORACLE_HOME/rdbms/lib/config.c file database group to match your UNIX group and re-compiled the Oracle binaries.
As specified in Oracle note 400459.1, you should move the old $ORACLE_HOME/rdbms/lib/config.o to config.o.old and then re-link oracle using “relink oracle”.
Check that a new config.o file is generated, before performing the “relink all”.