Let $X$ be a 1-connected compact space such that the algebra $H(X; \mathbb{F}_2)$ is generated by one single element. We compute the cohomology of the free loop space $H(\Lambda X; \mathbb{F}_2)$ including the Steenrod algebra action. When $X$ is a projective space $\mathbb{C}P_n$, $\mathbb{H}P_n$, the Cayley projective plane $\mathbf{Ca}P^2$ or a sphere $S^m$ we obtain a splitting result for integral and mod two cohomology of the suspension spectrum $\Sigma^\infty(\Lambda X)_+$. The splitting is in terms of $\Sigma^\infty X_+$ and the Thom spaces $Th(q \tau )$, $q \geq 0$ of the $q$-fold Whitney sums of the tangent bundle $\tau$ over $X$.