/usr/lib/x86_64-linux-gnu/libQtDesigner.so.4 is in libqt4-designer 4:4.8.7+dfsg-7ubuntu1.
This file is owned by root:root, with mode 0o777.
It is a symlink to /usr/lib/x86_64-linux-gnu/libQtDesigner.so.4.8.7
/usr/lib/x86_64-linux-gnu/libQtDesigner.so.4.8.7