  We determine the mod $2$ cohomology over the Steenrod algebra of the
classifying spaces of the free loop groups $LG$ for compact groups $G=Spin(7)$,
$Spin(8)$, $Spin(9)$, and $F_4$. Then, we show that they are isomorphic as
algebras over the Steenrod algebra to the mod $2$ cohomology of the
corresponding Chevalley groups of type $G(q)$, where $q$ is an odd prime power.
In a similar manner, we compute the cohomology of the free loop space over
$BDI(4)$ and show that it is isomorphic to that of $BSol(q)$ as algebras over
the Steenrod algebra.
