Hall and hyperfocal implies retract

Statement
Suppose $$G$$ is a finite group and $$H$$ is a uses property satisfaction of::Hall subgroup of $$G$$ that is also a  hyperfocal subgroup: its focal series in $$G$$ terminates at the trivial subgroup. Then, $$H$$ is a proves property satisfaction of::retract (and in particular a  proves property satisfaction of::Hall retract) of $$G$$, i.e., it possesses a  normal complement.