MAKE SURE YOU HAVE A BACKUP !!!
LOLLIPOP is new and thus not bug free !!!
1.1 You need obviously an already rooted phone with [NUT]'s DualRecovery installed and working (this can be achieved on firmwares with Android 4.4.2.)
If you don't have the recovery, you can install it from here: [Z2][TWRP][PhilZ Touch][CWM] XZDualRecovery Q&A see post #2 there for instructions. 2.8.12 and later should work on KitKat and Lollipop too!
1.2 You need Flashtool with working USB drivers (Make sure flashmode works - which means the drivers are properly installed on your system - before you start this tutorial).
2. Download the new FTF update file with 4.4.4/5.0.2 for your phone/branding. Use the same customization version (xxxx-xxxx) that is on your phone now!
This you can do with Flashtool (Devices/Check Updates menu) or download it from the [FTF] STOCK FIRMWARES for Xperia Z2/Z2a (D65x3/D6502/SO-03F) thread.
(Alternative option for advanced users: XperiFirm -> you can create an FTF from the downloaded firmware files with Flashtool.)
3.1 Download PRFCreator v0.5.zip from [TOOL] PRFCreator - easily create pre-rooted firmwares. V0.7/0.8 may work, but I couldn't test it. You are always safe with v0.5.
3.2 Download latest stable SuperSU update package from here.
3.3 Download XZDualRecovery's latest - 2.8.14 - flashable version (Z2-lockeddualrecovery2.8.x-RELEASE.flashable.zip) from here (XZDualRecovery section).
23.0.1.A.0.167,23.0.1.A.3.9,23.0.1.A.4.30,23.0.1.A .4.44,23.0.1.A.5.77,23.0.A.2.105,23.0.A.2.106,23.0 .A.2.93,23.0.A.2.98,23.0.F.1.74
You will have 5 files so far:
D650x_23.x.1.A.x.xxx_Telekom_HU.ftf (with different name and firmware version number)
5. Load the FTF, SuperSU and Z2-lockeddualrecovery (flashable) zip files into the three section.
6. Uncheck all the Include options [ ], we don't need them, we will flash them with Flashtool after the update. I think it's still safer to flash kernel, baseband, boot and other system critical parts of the firmware from Flashtool.
(But if you want to include them anyway then you can skip part 11 and start your phone normally.)
7. Push Create. After it's finished (wait for the Create button to be available again), you'll find a flashable.zip in the folder of PRFCreator.
Check the contents of the zip file, it should look like this with PRFCreator v0.5:
META-INF busybox system.ext4 dualrecovery.zip SuperSU.zip
8. Copy flashable.zip to the internal/external SD card of your phone.
9. Boot your phone into recovery (TWRP recommended).
10. Install flashable.zip in recovery. After finished DO NOT REBOOT !!! Clear Cache/Dalvik only and go back to Home and choose Reboot/Power off.
11. Start Flashtool and with the Flash device icon flash the kernel and other baseband files from the FTF in Flashmode.
Use the following configuration (or similar):
[ ] APPSLOG [ ] CACHE [ ] DATA
[ ] BOOTBUNDLE [x] PARTITION [x] TA -> not always present in the FTF file, but if you see it, then put an x here, just to be sure, you don't simlock your device [ ] KERNEL [ ] FOTA [ ] BASEBAND [x] SYSTEM [ ] ELABEL
[x] No final verification [ ] Reset customization
Push Flash button.
Push the Volume down button and connect the USB cable to the computer's USB port. Wait until Flashtool begins with the flashing and you can release the Volume button now.
When finished, unplug the the phone, start it and wait until it completes the first boot. After it is fully working, check SuperSU and if you can mount system R/W too (in Root Explorer or in other root file manager).
ENJOY another rooted version of Android I can give support only for this process, for the tools you need to ask your questions in the appropriate threads above.
Thanks to @zxz0O0 for creating the PRFCreator and the Lollipop fixes, it makes things easier in this process (you don't need to mess with linux mounts, virtual machines and other scripts copying system images and recovery updates to a zip).