  In this paper, we mainly investigate the converse of a well-known theorem
proved by P. Hall, and present detailed characterizations under the various
assumptions of the existence of some families of Hall subgroups. In particular,
we prove that if $p\neq 3$ and a finite group $G$ has a Hall $\{p,q\}$-subgroup
for every prime $q\neq p$, then $G$ is $p$-soluble.
