Almost fixed points of finite group actions on manifolds without odd cohomology

If $X$ is a smooth manifold and ${\mathcal{G}}$ is a subgroup of $Diff(X)$ we say that $(X,{\mathcal{G}})$ has the almost fixed point property if there exists a number $C$ such that for any finite subgroup $G\leq{\mathcal{G}}$ there is some $x\in X$ whose stabilizer $G_x\leq G$ satisfies $[G:G_x]\leq C$. We say that $X$ has no odd cohomology if its integral cohomology is torsion free and supported in even degrees. We prove that if $X$ is compact and possibly with boundary and has no odd cohomology then $(X,Diff(X))$ has the almost fixed point property. Combining this with a result of Petrie and Randall we conclude that if $Z$ is a non necessarily compact smooth real affine variety, and $Z$ has no odd cohomology, then $(Z,Aut(Z))$ has the almost fixed point property, where $Aut(Z)$ is the group of algebraic automorphisms of $Z$ lifting the identity on $Spec\,{\mathbb{R}}$.