Is it possible to find a torsion-free space with non-trivial $\operatorname{Sq}^{2^n}$ ?
Torsion-free means integral cohomology is torsion-free. When $n \leq 3$, such spaces are projective space $\mathbb{CP}^n$, $\mathbb{HP}^n$ and the Cayley plane. $\operatorname{Sq}^{2^n}$ takes the generator of degree $2^n$ to the generator of degree $2^{n+1}$. So what is such a space for $n > 3$? That is, can we find a torsion-free space $X$ such that the indecomposable element $\operatorname{Sq}^{2^n}$ acts non-trivially on mod 2 cohomology of $X$?
$\textrm{Sq}^{2^n}$ acts non-trivially on the mod 2 cohomology of $\mathbb{C}P^{m}$ if $m \geq 2^{n}$: it takes the nonzero element in dimension $2^n$ to the nonzero element in dimension $2^{n+1}$. This is true because for any cohomology class $x$ in dimension $d$, $\textrm{Sq}^d x = x^2$.
If you want a space which has nonzero cohomology only in dimensions $2^n$ and $2^{n+1}$, connected by $\textrm{Sq}^{2^n}$, this is impossible for $n\geq 4$ by Adams' Hopf invariant one theorem