In the setting of finite groups, suppose $J$ acts on $N$ via automorphisms so that the induced semidirect product $N\rtimes J$ acts on some non-empty set $\Omega$, with $N$ acting transitively. Glauberman proved that if the orders of $J$ and $N$ are coprime, then $J$ fixes a point in $\Omega$. We consider the non-coprime case and show that if $N$ is abelian and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$. We also show that if $N$ is nilpotent, $N\rtimes J$ is supersoluble, and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$.

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.