For fixed $C \in \mathrm{\mathbf{GL}}_n(\mathbb{R})$, define $\varphi, \psi: \mathrm{\mathbf{SL}}_n(\mathbb{R}) \rightarrow \mathrm{\mathbf{SL}}_n(\mathbb{R})$ as $$\varphi(A) = CAC^{-1}, \quad \psi(A) = (A^{\mathrm{\mathbf{t}}})^{-1}, \qquad (A\in \mathrm{\mathbf{SL}}_n(\mathbb{R}))$$
I've shown that both are automorphisms of ${\mathbf{SL}}_n(\mathbb{R})$, and I've shown that for C having positive determinant or n odd, $\varphi$ is an inner automorphism.
But I can't quite grasp for the case when $n$ would be even. Some reference I looked at stated it's not necessarily inner for $n$ even, but I'm looking for precisely when it is and when it is not.