1 - Download the OEM 12c R2. It can be downloaded here:

http://www.oracle.com/technetwork/oem/grid-control/downloads/index.html

 Note: it is a set of three huge zips.

2 - Unzip the archives

3 - Create a directory (as the oem-owner user) where the upgraded Middleware should be installed. For instance:

$ mkdir /u01/app/oracle/Middleware12cR2

4 - Back up OMS (Middleware home and inventory), Management Repository and Software Library.

http://docs.oracle.com/cd/E24628_01/doc.121/e24473/ha_backup_recover.htm#EMADM10740

5 - Ensure that the Management tables don't have snapshots:

 SQL> select master , log_table from all_mview_logs where log_owner='<EM_REPOS_USER>

If there are snapshots drop them:

 SQL> Drop snapshot log on <master>

6 - Copy emkey  from existing OMS:

 $ <OMS_HOME>/bin/emctl config emkey -copy_to_repos [-sysman_pwd <sysman_pwd>]

To verify whether the emkey is copied, run the following command:

 $ <OMS_HOME>/bin/emctl status emkey

If the emkey is copied, then you will see the following message:

The EMKey  is configured properly, but is not secure.
Secure the EMKey by running "emctl config emkey -remove_from_repos".

7 - Stop the OMS and the Agent

$ <OMS_HOME>/bin/emctl stop oms

$ <AGENT_HOME>/bin/emctl stop agent

8 - from the unzipped directory, run

$ ./runInstaller

8a - Follow the wizard: Email / MOS; Software Updates: disable or leave empty.

8b - Follow the wizard:  Installation type: Upgrade -> One System Upgrade.

8c - Installation Details: Middleware home location: enter the directory created in step 3.

8d - Enter the DB Connections Details. Credentials for SYS and SYSMAN.

8e - Dialog comes: Stop the Job Gathering: click 'Yes'.

8f- Warning comes: click 'OK'.

8g - Select the plugins to deploy along with the upgrade process

8h- Extend Weblogic: enter the password (recommended, the same password for the SYSMAN user).

A new directory will be created, recommended:

/u01/app/oracle/Middleware12cR2/gc_inst

8i - Let the upgrade proceed by clicking 'Install'.

8j - Run the following script (as root) and finish the 'installation':

 $ /u01/app/oracle/Middleware12R2/oms/allroot.sh

9 - Turn on the Agent:

 $ <AGENT_HOME>/bin/emctl start agent

 Note that the $AGENT_HOME might be located in the old Middleware directory:

 $ /u01/app/oracle/Middleware/agent/agent_inst/bin/emctl start agent

10 - go to the EM UI. Select the WebLogic Target and choose the option "Refresh WebLogic Domain" from the menu.

11 - Update the Agents: Setup -> Manage Cloud Control -> Upgrade Agents -> Add (+)

Note that the agents may take long to show up.

... and that's it! Or that should be it !