Rename command not found when running a script

I just had to install the rename package.

sudo apt install rename

My Ubuntu has a man page for rename(1) on fresh install, but running rename gives a CNF error.

The man page style suggests that the command belongs to GNU Core Utilities, but confusingly it actually is rename.ul (Ubuntu Bionic / Disco / Eoan + Debian Buster). This command corresponds to the mysterious rename(1) page.

If you install rename package, the new man page is rename(1p). It's a Perl script, not coreutils.