Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(-k)\right\} \\\subseteq \left\{ f: (I^d,\partial I^d)\to (X,x_0) \right\} = \Omega^dX,$$ where $I = [-1,1]$.

Equip the sets $S_d$ and $\Omega^dX$ with the Compact-open topology, such that they become topological spaces. What can we say about the homotopy groups $\pi_n(S_d,c_{x_0})$, where $c_{x_0}$ is the constant map into $x_0$?

I am looking for a strategy in computing $\pi_n(S_d,c_{x_0})$. What I do know are the homotopy groups $\pi_n(\Omega^dX,c_{x_0})\cong \pi_{n+d}(X,x_0)$, which is a standard result in homotopy theory. But $S_d$ is a subspace in $\Omega^dX$, which does not have to share the same homotopy groups. The elements of $S_d$ satisfy a certain $\mathbb{Z}_2$-equivariance condition and the theory about $G$-equivariant homotopy seems to be very involved, although I would certainly dive into it, when I knew that there were tools with which one could calculate the homotopy groups of $S_d$.

Thank you in advance!

Edit: Consider as an example $X = V_p(\mathbb{C}^q)$, the complex Stiefel manifold, whose elements we will interpret as complex $q\times p$-matrices.