GNU screen quitting -- sigterm to running process

GNU screen quitting -- sigterm to running process

I want to quit screen session. For this, I use screen -S session_name -X
quit. This works well, but not like I would.
The program running in screen has a handler for SIGTERM, and I really need
this handler to execute, so that it can perform proper cleanup. However,
running screen -S session_name -X quit results in the program quitting
without having its handler invoked.
A simple kill $(pidof bla) results in the invocation of my handler, proper
cleaning, and finally the closing of the screen session.
However, I'd like to be able to stop all this by closing the screen using
its session's name, and letting it "forward" the SIGTERM signal.
Any help appreciated, thanks.