  We demonstrate that two supersoluble complements of an abelian base in a
finite split extension are conjugate if and only if, for each prime $p$, a
Sylow $p$-subgroup of one complement is conjugate to a Sylow $p$-subgroup of
the other. As a corollary, we find that any two supersoluble complements of an
abelian subgroup $N$ in a finite split extension $G$ are conjugate if and only
if, for each prime $p$, there exists a Sylow $p$-subgroup $S$ of $G$ such that
any two complements of $S\cap N$ in $S$ are conjugate in $G$. In particular,
restricting to supersoluble groups allows us to ease D. G. Higman's stipulation
that the complements of $S\cap N$ in $S$ be conjugate within $S$. We then
consider group actions and obtain a fixed point result for non-coprime actions
analogous to Glauberman's lemma.
