/usr/bin/designer-qt4 is in qt4-designer 4:4.8.5+git192-g085f851+dfsg-2ubuntu4.1.
This file is owned by root:root, with mode 0o777.
It is a symlink to /usr/bin/../lib/x86_64-linux-gnu/qt4/bin/designer
/usr/bin/../lib/x86_64-linux-gnu/qt4/bin/designer