During my research I have recently stumbled upon the problem of finding the relative homotopy sets $\pi_1(U_n,U_n/O_n)$ and $\pi_1(U_{2n},U_{2n}/USp_{2n})$ for $n$ large enough to be in the stable regime where Bott periodicity applies.

The subgroup $U_n/O_n\subset U_n$ contains all elements $g\in U_n$ with $g=g^T$ and the Cartan embedding is \begin{align} U_n/O_n&\hookrightarrow U_n\\\ g\cdot O_n&\mapsto gg^T \end{align} Similarly, the subgroup $U_{2n}/USp_{2n}\subset U_{2n}$ contains all elements $g\in U_{2n}$ with $g=Jg^TJ^{-1}$ and the Cartan embedding is \begin{align} U_{2n}/USp_{2n}&\hookrightarrow U_{2n}\\\ g\cdot USp_{2n}&\mapsto gJg^TJ^{-1}, \end{align} with $J:=\begin{pmatrix}0&1_n\newline-1_n&0\end{pmatrix}$.

In order for my other results to be consistent, I expect the result to be
\begin{align}
X:=\pi_1(U_n,U_n/O_n)&=0\\\
Y:=\pi_1(U_{2n},U_{2n}/USp_{2n})&=\mathbb{Z}_2
\end{align}
However, with the following argument I find that *both* are trivial: Using the exact sequence of homotopy groups/sets,
\begin{align}
\pi_1(U_n/O_n)&\stackrel{i_1}{\to}\pi_1(U_n)\to X\to\pi_0(U_n/O_n)\\\
\pi_1(U_{2n}/USp_{2n})&\stackrel{i_2}{\to}\pi_1(U_{2n})\to Y\to\pi_0(U_{2n}/USp_{2n}),
\end{align}
where $i_{1}$ and $i_2$ are induced by the Cartan embeddings above, there is the bijection
\begin{align}
X=\pi_1(U_n)/\text{img}(i_1)=\mathbb{Z}/\text{img}(i_1)\\\
Y=\pi_1(U_{2n})/\text{img}(i_2)=\mathbb{Z}/\text{img}(i_2)
\end{align}
since $\pi_0(U_n/O_n)=\pi_0(U_{2n}/USp_{2n})=0$.

The determinant induces a bijection $\pi_1(U_n)\to\pi_1(U_1)=\mathbb{Z}$ with inverse induced by $U_1\hookrightarrow U_n$ (adding ones to the diagonal), so a representative loop of the class $m\in\pi_1(U_n)=\mathbb{Z}$ is $$\phi:t\mapsto\text{diag}(e^{i2\pi m t},1,\dots,1).$$ This has a preimage under $i_1$, namely the loop $$\phi_1:t\mapsto\text{diag}(e^{i\pi m t},1,\dots,1)\cdot O_n.$$ This is indeed a loop since $\phi_1(0)=(1,1,\dots,1)\cdot O_n$ and $\phi_1(1)=(-1,1,\dots,1)\cdot O_n=(1,1,\dots,1)\cdot O_n$.

Similarly, there is a preimage under $i_2$, namely the loop $$\phi_2:t\mapsto\text{diag}(e^{i\pi m t},1,\dots,1)\cdot USp_{2n}.$$ Again, this is a loop since $\phi_2(0)=(1,1,\dots,1)\cdot USp_{2n}$ and $\phi_2(1)=(-1,1,\dots,1)\cdot USp_{2n}=(1,1,\dots,1)\cdot USp_{2n}$. Thus, both $i_1$ and $i_2$ are surjective and $X=Y=0$.

However, for the expected result to hold, $i_1$ should be surjective in the sequence with $X$ and $\text{img}(i_2)=2\mathbb{Z}$ (even integers only) in the sequence with $Y$. This does not seem to be the case - is my argument flawed or is it true that both relative homotopy sets are in fact trivial?

Thank you!