Unable use F1 to F4 key mappings with urxvt

I am, just starting to use Kakoune, which I find a great editor.
I am on Debian,GNU/Linux, kakoune v. 2019.01.20.

The problems happens when I use kak in a urxvt terminal, while not with gnome-terminal.
I have no problem to map function keys form F5 to F12, but when I try to use mappings of, e.g., F2 I get, if in insert mode, 12~ printed. (Typing in the terminal cat F2 yields ^[[12~ ). This happens for functions keys F1, F2, F3 and F4.

Thanks for any suggestion

What is $TERM set to? Inside gnome-terminal, it should be vte-256color (or possibly xterm-256color), while inside urxvt it should be rxvt-unicode-256color.

Most modern terminals try to behave the same way as xterm, and so set $TERM to xterm or xterm-256color for compatibility. urxvt is one of the few modern terminals that is deliberately incompatible with xterm, and one of the biggest differences is in the way it handles function keys, so make sure $TERM is set to some kind of rxvt-related name.

This is the current setting, both for urxvt and gnome-terminal:
echo $TERM

As you suggest, setting TERM=urxvt-unicode-256color solves the problem.