Fixed point conditions for non-coprime actions

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