= Solution
The <matrix exponential> is smooth, and part (a) identifies its derivative at zero as the identity. The <inverse function theorem> therefore gives a local <diffeomorphism> from an open neighborhood $U_0$ of zero onto a neighborhood of $I$. Choose a sufficiently small <operator norm> ball $U\subset U_0$ and let $V=\exp U$. This is again a <diffeomorphism> onto an open set, since it is a restriction of the original one.
The ball $U$ is preserved by $A\mapsto A^T$ and $A\mapsto-A$. Since $(\exp A)^T=\exp(A^T)$ and $(\exp A)^{-1}=\exp(-A)$, $V$ is preserved by adjoint and inversion. For its local inverse $L$, injectivity also gives
$$
\boxed{L(Q^T)=L(Q)^T,\qquad L(Q^{-1})=-L(Q)}\qquad(Q\in V).
$$
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