Yes, it is permissible and Ok to pray The Tahajjud half an hour before FajR, Tahajjud can be at anytime between Ishaa and Fajr, and this is considered part of the night prayer as long as the time of Fajr did not enter.
The Tahajjud takes place in the last third of the night, and whoever prays The Tahajjud before Fajr, there is nothing wrong with that because the night continues until the time of Fajr. .
