= Solution
Near $t=0$, take $\Omega(t)=\log F(t)$ to be the continuous local <matrix logarithm> with $\Omega(0)=0$. The symmetry identity gives $F(-t)=F(t)^{-1}$, and uniqueness of this logarithm yields
$$
\Omega(-t)=\log(F(t)^{-1})=-\log F(t)=-\Omega(t).
$$
Thus $\Omega$ is an <odd function>. The local-logarithm qualification is necessary because the <matrix exponential> is not globally injective; the statement is naturally understood either near $t=0$ or as an identity of formal power series.
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