  The rational Borel equivariant cohomology for actions of a compact connected
Lie group is determined by restriction of the action to a maximal torus. We
show that a similar reduction holds for any compact Lie group $G$ when there is
a closed subgroup $K$ such that the cohomology of the classifying space $BK$ is
free over the cohomology of $BG$ for field coefficients. We study the
particular case when $G$ is a semi-direct product and $K$ is its maximal
elementary abelian 2-subgroup for cohomology with coefficients in a field of
characteristic two. This provides a different approach to investigate the
syzygy order of the equivariant cohomology of a space with a torus action and a
compatible involution, and we relate this description with results for 2-torus
actions.
