  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 $G/H$.
