Topological group actions by group automorphisms and Banach representations

We study Banach representability for actions of topological groups on groups by automorphisms (in particular, actions of groups on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces. In contrast to the standard left action of a locally compact second countable group $G$ on itself, the conjugation action need not be reflexively representable even for $\mathrm{SL}_2(\mathbb{R})$. The conjugation action of $\mathrm{SL}_n(\mathbb{R})$ is not Asplund representable for every $n \geq 4$. The linear action of $\mathrm{GL}(n,\mathbb{R})$ on $\mathbb{R}^n$, for every $n \geq 2$, is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of $l_1$). The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). This is unclear for $\mathrm{SL}_2(\mathbb{R})$. As a byproduct we obtain some counterexamples about Banach representations of homogeneous $G$-actions $G/H$.